Relativity

Contents

Classical Mechanics

Classical mechanics describes the way objects move and interact in accordance with Newton's laws of motion. The basic assumptions involve a frame of reference (x,y,z) with respect to which object with mass m moves, there is an independent time variable t to record the sequence of the movement, the gravitational or electromagnetic interaction between objects is instantaneous, and objects with geometric extent are often idealized as a point (with the justification that the size is much smaller than the distance involved). The basic equation is:

Figure 01a Newton's 3 Laws [view large image]

where G is the gravitational constant, m1 and m2 are the masses of the two objects interacting via gravitation, r is the distance between these two objects, and (r/r) is an unit vector along the direction of r (see Figure 01b). If one of the objects is much heavier than the other, e.g., m1 >> m2 like the Sun / Earth system, then m1 can be placed in the origin of the coordinate system and Eq.(1) can be solved as a one-body problem. In case the two masses are similar, the problem can be reduced to a one-body problem with a fictitious object moving around the center of mass, and Eq.(1) is still applicable. The equation of motion becomes rapidly un-manageable for system of three bodies and beyond. Eq.(1) would include accelerations for all the objects and the force on one object would involve the interaction with all the others. This is the situation often encountered in celestial mechanics with spacecraft flying among planets. The

Figure 01b Gravitational Interaction [view large image]

solution is usually obtained by some kind of approximation and by numerical computation using large computers. See Newton's Laws in cartoons.

dl2 = dx2 + dy2 + dz2 = dx'2 + dy'2 + dz'2 ---------- (5) According to Eq.(4) if the velocity of light (in the x direction) is c in the S frame it would be c' = c - V in the S' (moving) frame. In classical mechanics, the "Absolute Frame of Reference" is a hypothetical entity identified as the frame of reference with the origin at the center of mass of system of fixed stars. Only with respect to this absolute frame of reference would the velocity of light equal to c = 3x1010 cm/sec.

Figure 02a Galilean Transformation [view large image]

It was further suggested that the medium in which light propagates - the ether - would be an even better absoute frame of reference.

Special Relativity

The search was on in the late 19th Century for this elusive ether. Finally in 1887 Michelson and Morley demonstrated conclusively that the speed of light in different inertial frames is the same everywhere (Figure 02b). The Maxwell's equations in electro-magnetism also indicates that the speed of light is a constant regardless of the relative motion of the person measuring that speed. But as just mentioned above, the velocity of light is different in different inertial frame according to the Galilean transformation. The theory of special relativity was postulated to reconcile this kind of inconsistency. The theory does away with the idea of absolute frame of reference such as the ether, and treats all inertial frames on an equal footing.

Figure 02b Michelson-Morley Experiment [view large image]

	Mathematically if a spherical light wave is generated at the origin of the S and S' inertial frames when they are coincided at t = 0, the statement about the constant velocity of light in different inertial frames can be expressed as: x2 + y2 + z2 = c2 t2     (for an observer in the S frame) ---------- (6) x'2 + y'2 + z'2 = c2 t'2     (for an observer in the S' frame) ---------- (7)
Figure 02c Constant Speed of Light [view large image]

• The independent time variable t is now treated on the same footing as the other spatial dimensions.




The length L' = x'2 - x'1 in the moving inertial frame is now different to the length L with respect to a stationary observer in the S system at t = to. According to the first formula in Eq.(8a), the relation is given by: L = x2 - x1 = L' (1 - V2/c2)1/2, which is known as the Lorentz contraction. The length of a moving rod is shorter according to a stationary observer. The phenomena is demonstrated in high energy collision, when the round shape of atom becomes a pancake with the flatten face perpendicular to the direction of motion (see Figure 03).

Figure 03 Lorentz Contraction[view large image]

• The time interval T' = t'₂ - t'₁ of a clock sitting in the S' system at x' = x'_o appears to take longer to tick with respect to a stationary observer in the S system. According to the 4^th formula in Eq.(8b), the relation is given by T = t₂ - t₁ =
  T' / (1 - V²/c²)^1/2. The effect is known as time dilation and T' is referred to as the proper time - the time experienced in the clock's rest frame. The phenomenon is demonstrated in the decay of unstable particle moving at near the speed of light. The lifetime of such particle appears to be much longer than the one measured in a stationary lab. The diagrams in Figure 04a (where = (1 - V²/c²)^-1/2), show the muon decay as experienced in its own frame and from the view point

  		of a stationary observer on Earth. The twin paradox is another consequence of time dilation. As shown in Figure 04b, one of the twin (at O) leaves on a space journey during which he travels close to the speed of light, while his sister remains on Earth. Because of his motion, time runs more slowly in the spacecraft as seen by the earthbound twin. So on his return the space traveler (at B) will find that his sister has aged more than himself as indicated by the clock readings in line OB (for the sister) and OAB (for the brother). The paradox arises because it can be argued that the sister is moving near the speed of light relative to her brother and so the brother should be getting older instead. A number of experiments have confirmed that the traveling twin would indeed be younger. The two world-
  Figure 04a Dilation [view large image]	Figure 04b Twin Paradox [view large image]	lines are different, and not interchangeable, as there is no inertial frame in which the traveling twin is always at rest.

  Events (a point in the four dimensional space-time as shown in Figure 04c) that take place simultaneously to an observer in S, e.g., t = to at x = x1 and x = x2, are separated by a time interval t'2 - t'1 = (V(x1 - x2)/c2) / (1 - V2/c2)1/2 in S' according to the fourth formula in Eq.(8a) (see Figure 05). For an infinitesimal interval of space-time, Eq.(6) can be written in the form: c2 dt2 - dx2 - dy2 -dz2 = 0 ---------- (9) This is called the null line for object moving at the speed of light. In place of the

  Figure 04c Minkowski Space- time

  invariant lenght dl in Eq.(5) under the Galilean Transformation, it can be shown that for object moving below the speed of light, the space-time interval (in real number)

  ds² = c² dt² - dx² - dy² - dz² ---------- (10)
  
  is invariant under the Lorentz transformation. ds is called the proper time since it is the time interval for a clock at rest in a reference frame with dx = dy = dz = 0. In general, denoting v² = [(dx/dt)² + (dy/dt)² + (dz/dt)²], Eq.(10) can be re-written into the form: ds² = (1-v²/c²)c²dt². The interval is called time-like if ds² 0 for v c. As shown in figure 04c,

  event 2 can be related causally (temporal sequence cannot be reversed) in some way to event 1 provided that a signal (traveling slower than the speed of light) is available. If ds2 < 0, then it is called space-like, which implies Event 3 is entirely unrelated to Event 1. Alternatively it can be interpreted that two events joined by a space-like interval can never influence each other, since that would imply a flow of information at speeds faster than the speed of light. For some space-like interval, e.g., to event 4, the signal signifies backward in time as well. The interval ds plays the role of the time parameter in Newtonian mechanics to keep track of the development of events such as in the generalized equation of motion in Eq.(12b).

  Figure 05 Minkowski Spacetime Transformation [large image]

  Note that the velocity of light c is constant in all Lorentz frame of references in Figure 05 as originally envisioned.

• Closer examination reveals that Eq.(10) is not a sum of squares of the coordinate differentials. One of these is associated with an opposite sign. This is called pseudo-Euclidean geometry and is the reason for the un-usual appearance of the space-time axes when subject to a rotation in the Minkowski space-time as shown in Figure 05. Mathematically, the Minkowski space-time rotation can be expressed in a form resembling to the 2-dimensional rotation by re-writing Eq.(8b) in term of the hyperbolic functions:
  
  x = x' cosh(A) + c t' sinh(A)
  ct = x' sinh(A) + c t' cosh(A)
  
  where tanh(A) = (V/c), sinh(A) = (V/c) , cosh(A) = are the hyperbolic functions, and = 1 / (1 - V² / c²)^1/2. As V approaches c, tanh(A) ~ 1, the x' and ct' axes merge together at the null line (see Figure 05).
• In Galilean transformation, the velocity with respect to the S' system is v' = v -V, which is now replaced by
  v' = (v - V) / (1 - vV/c²) in special relativity. Incidentally, this formula yields correctly v' -c as V c.
• Using the proper time ds as the parameter independent of different inertial frames, the velocity in the 4 dimensional space-time can be defined as:
  u = dx/ds = v/c
  where the Greek indices run from 1 to 3 representing x, y, z, and the 4th component
  u₄ = icdt/ds = i.
  By definition the square of the 4-velocity u² = -1 is invariant under the Lorentz transformation.
• Similarly the 4-momentum is defined as:
  p = m_ov,
  and p₄ = im_oc.
  The square of the 4-momentum p² = -m_o²c² is another 4-scalar quantity invariant under the Lorentz transformation.
  The 4th component is linked to the total energy E by the formula:
  E = ip₄c = m_oc² = mc²
  where m = m_o. The relationship is justified by the correct limit at low speed when E m_oc² + m_ov²/2. This is the origin of the most celebrated formula invented by Einstein.
• Thus, the mass for a particle moving at velocity v is given by m = m_o / (1 - v²/c²)^1/2, where m_o is the rest mass (relative to the observer's rest frame). This formula shows that m as v c. Thus objects with mass can never be accelerated to the speed of light or greater.
• It also shows that mass and energy are equivalent. They can be converted to each other. The total energy E of a particle is given by the formula E = m c² or E = m_o c² + T. The first term is the rest mass energy, and the second term is the kinetic energy (at speed up to c) T = [1/(1 - v²/c²)^1/2 -1] m_oc².
• Now in term of the 3-momentum p, and E = (m_o² c⁴ + p² c²)^1/2. The three-momentum and energy together form a 4-vector (p_x, p_y, p_z, iE/c) in the Minkowski space. Its magnitude m_o² c⁴ is invariant under the Lorentz transformation.

From the relations between E, m, and p, the velocity v can be expressed in term of the 3-momentum p: (v/c)2 = 1/[1 + (moc/p)2]. For mo2 > 0, the particle is always moving slower than the velocity of light. For mo2 = 0, v is always equal to c. If mo2 < 0 then v > c, i.e., the particle is moving faster than the velocity of light. Such particle is called tachyon, which has the peculiar property that it slows down to approach the velocity of light with increasing momentum p, while its speed goes up to infinity as p falls to |mo|c (see Figure 06).

Figure 06 Relativistic Velocity [view large image]

There is no direct evidence that tachyons exist, and most physicists believe there is something wrong or it requires some sort of interpretation when they appear in a theory. The spontaneous symmetry breaking potential in the Higgs mechanism is one

example of tachyon in physics. Note that the Higgs particle is not tachyonic, it acquires real mass after symmetry breaking from the unstable configuration. Another example is the quantized bonsonic string, which admits tachyonic state. It is eliminated after introducing super-symmetry into the theory.
• Table 01 illustrates the differences between timelike and spacelike objects, where x and k = p denote the 3-D spacetime vector and 3-D momentum respectively, and v = |dx/dt|. In quantum field theory, the virtual particles possess spacelike or off mass-shell characteristics, which enable physicists to resolve a variety of problems. Off mass-shell means that the relation m_o²c² = E²/c² - k² is no longer valid. For a real photon ds = 0, and m_o = 0, otherwise it is an off mass-shell virtual photon.
  
  

  Mathematical Entity	Timelike	Spacelike	Spacelike Property
  4-D Spacetime Vector	x = (x, ict)	x = (x, ict)	Outside light cone
  Spacetime Interval	ds2 = c2dt2 - dx2 0	ds2 = c2dt2 - dx2 < 0	v > c - tachyon
  4-Momentum	k = (k, iE/c)	k = (k, iE/c)	Outside light cone
  4-Momentum Squared	mo2c2 = k2 = E2/c2 - k2 0	mo2c2 = k2 = E2/c2 - k2 < 0	Off mass-shell - tachyon
  4-Momentum Transfer	k' - k = [(k' - k), i(E' - E)/c] = q	k' - k = [(k' - k), i(E' - E)/c] = q	Outside light cone
  Momentum Transfer Squared	(k' - k)2 = (E' - E)2/c2 - (k' - k)2 = q2 0	(k' - k)2 = (E' - E)2/c2 - (k' - k)2 = q2 < 0	Off mass-shell

  Table 01 Spacelike Characteristics

  A general rule for space-like quantity is for those 4-scalar such as ds, m_o, or q to become an imaginary number.
• In special relativity the equation of motion for a particle is:
  
  f_i = m_oc² (du_i/ds)
  
  where the space component of the velocity is u = (v/c)/(1-v²/c²)^1/2, the time component is u₄ = i /(1-v²/c²)^1/2, while the space components of the four vector f_i form a vector f / (1-v²/c²)^1/2, the time component is f₄ = i f(v/c) / (1-v²/c²)^1/2. This equation is invariant under the Lorentz transformation. In general, all laws of physics are required to be invariant under the Lorentz transformation. It means that all physical laws are prescribed to retain the same form in all inertial frames of reference. This is the "principle of general covariance" originally envisioned by Einstein for special relativity. It has since then been generalized to encompass many different kinds of transformation such as the coordinate transformation and gauge transformation, ... etc.

General Relativity

		The inertial frames of reference in both classical mechanics and special relativity move with a constant velocity related to each others. Such arrangement seems to become impossible in the presence of gravity, which produces acceleration (change of velocity). However, there is a class of inertial frames of reference that can be obtained locally by letting it freely falling. This kind of frames would generate an opposite force, which exactly nullifies the acting force. The local region (such as in an
Figure 07a Free-falling Frame [view large image]	Figure 07b Equivalence Principle [view large image]	elevator) would experience zero gravity as shown in Figure 07a. Figure 07b shows the similar kind of situation in producing gravity with acceleration. The inter-changeable nature of gravity and acceleration (at least locally) is known as the principle of equivalence.

ds2 = gik dxi dxk ---------- (11) where the notations have been simplified such that x = x1, y = x2, z = x3, ct = x4; the indices i, k run from 1 to 4 and the repeated dummy index in the equation is understood to be summed over the 4 space-time coordinates. The gik is known as space-time metric, which is a second rank tensor and a function of the space-time. For the inertial or free-falling frame (flat space-time), g11 = g22 = g33 = -1, g44 = 1, and gik = 0 for i k. Eq.(11) can be alternatively viewed as the space-time interval of a curved world-line (Figure 07c)

Figure 07c World-line [view large image]

as opposed to a straight world-line for the free field case. In general the space-time metric gik is determined by the nonlinear differential equations (see "Differential Equation" for a very brief introduction) as postulated by Einstein:

where

is a second rank tensor related to the curvature of space,

is the Christoffel symbol, and Tik is the energy-momentum tensor of matter-energy.

Using the gravitational field equation and the equation of motion, Einstein presented a calculation on the effect of GR on the advance of the perihelion of Mercury: = 6GM/(c2a(1 - e2)) ---------- (12c) where M is the mass of the Sun, a is the length of the semi-major axis, and e is the eccentricity of the ellipse. In Figure 08, the amount of the advance is greatly exaggerated. The actual advance due to the effect of GR is only 0.43 seconds of arc per year. The most recent and most accurate results seem to be converging towards a value that makes the GR predictions agree well with observation.

Figure 08 Perihelion Advance [view large image]

Schwarzschild's Solution and Black Hole

• As r , the space-time metric reduces to the expression for flat space-time:
  
  ds² = c²dt² - dr² - r² (sin² d² + d²) ---------- (14)
  
  
• At the non-relativistic limit where v²/c² << 1, an expression for the space-time metric can be derived from classical mechanics:
  
  ds² = (1 + 2V/c²) c²dt² - dr² - r² (sin² d² + d²)
  
  where V is the gravitational potential.
  Comparing with the space-time metric g₄₄ in Eq.(13) in the region where r > 2GM / c², the Newton's law of gravitation: V = - GM / r is recovered from general relativity. Thus, at the non-relativistic limit only the time dilation effect (relating to g₄₄) is important. The effect of spatial curvature (relating to g₁₁) is negligible at this limit.
  
  
• At the distance where r = r_s = 2GM/c², the escape velocity v_e = (2GM/r)^1/2 = c. It implies that even light cannot escape the grip of gravity at this point. This is Schwarzschild radius separating the black hole from the rest of the world.




		A black hole can form only when the Schwarzschild radius rs is outside the central object. For the Earth, and the Sun, rs is well inside the physical boundary (rs for the Earth is about 1 cm and for the Sun is about 3 km). Even the neutron star (with similar mass to the Sun) has a physical radius of about 10 km. Only collapsing stars or galactic center have the necessary condition to form a black hole. Figure 09a is an embedding diagram (Figure 09b) of a black hole. The two dimensional circles are the projection of three dimensional spheres - the
Figure 09a Black Hole [view large image]	Figure 09b Embedding Diagram [view large image]	hyperspace. The vertical axis denotes the "stretch" of space in the radial direction. The slope of the curve can be considered as representing the curvature of the space.

	Gravitational redshift is generated as the photons loss energy in overcoming the pull of the gravitational field (see Figure 09c). For the case of a black hole, the shifted wavelength is computed by the formula: = o / (1 - rs/r)1/2 At the Schwarzschild radius rs, the redshift becomes infinity. This is the effect that makes light invisible at rs.
Figure 09c Redshift [view large image]

• The gravitational field also induces time dilation as experienced by a far away stationary observer. It is in a similar form:
  
  T = T_o / (1 - r_s/r)^1/2
  
  where T_o is the time interval recorded by an observer plunging toward the black hole and T is the corresponding time perceived by a stationary observer. Thus, for the stationary observer it takes an infinite long interval for the other

  		observer approaching the Schwarzschild radius rs. However, the adventurer is not aware this curious effect on the time interval. His journey may be interrupted only by the tidal force, which is much more ferocious for black hole with smaller size (the tidal force at rs is equal to 2GM / rs3 = c6/(2GM)2, the + and - signs represent the stretch and squeeze parallel and perpendicular to the radial direction respectively, see Figure 09d). Figure 09e shows the time dilation for a space traveler located at a distance of 1.1 rs from the center of a black hole.
  Figure 09d Tidal Force	Figure 09e Time Dilation [view large image]

  Inside rs, all matter must fall towards the center, even light, which keeps coming from outside (but cannot get out from inside). After a finite time the inward falling object will end up at r = 0, where the density of matter is infinite. The metric g44 changes sign inside the black hole, i.e., time has become just another dimension of space. It is suggested such change has the effect of smoothly rounded off the singularity at r = 0. There is speculation that quantum effect will remove the singularity; it might provide a gateway leading to another universe. Figure 09f shows a collapsing star, and the behavior of light near and inside the black hole. The effect of the black hole on light is indicated by the direction of the light cone. It shows that far away from the black hole the light cone points to all directions around. The directions are tilted toward the black hole as it moves closer. At the event horizon it cannot escape to outside. The directions of the cone only point to the center once inside the black hole, there is no turning back.

  Figure 09f Black Hole, Inside [view large image]

  White holes are similar to black holes except white holes are ejecting matter while black holes are absorbing matter. The existence of white holes is implied by the negative square root solution to the Schwarzchild metric. In particular, if we define the proper time d to be the time associated with the moving frame, in which the spatial variation vanishes (co-moving objects are stationary in that frame), then

  Figure 09g White Hole [view large image]

  c2 d2 = ds2 = g44 c2 dt2 where g44 is positive according to the convention. Thus,

  d = ± (g₄₄)^1/2 dt
  where the positive sign corresponds to the black hole solution while the solution with negative sign is interpreted as the white hole with time running backward. Figure 09g depicts an embedding diagram of a white hole, which is just a black hole turned upside down to give an illusion that matter is spilling out instead of pouring in.



• The space-time metric for a static wormhole (also known as the Einstein-Rosen Bridge) can be expressed in the form:
  
  

  ---------- (15a)

  The lapse function defines the proper time between consecutive layers of spatial hyper-surfaces; while the shape function determines the shape of the worm hole. The shape function takes on a very simple form for the case of the Schwarzschild's metric, i.e., b(r) = 2GM / c2 = rs. The throat of the wormhole is located at r = b(r) = rs in this case. Figure 09h is a computer generated embedding diagram of a blackhole, a wormhole, and a whitehole. The surface of the diagram measures the curvature of space. Color scale represents rate at which idealized clocks measure time; red is slow, blue fast.

  Figure 09h Wormhole [view large image]

  Another way to conceptualize a wormhole topology is to have the spatial part of the space-time metric in Eq.(15a) imbedded in a flat hyperspace with the extra-dimension denoted by W:

  dl² = dW² + dr² + r² (sin² d²) = dr² / (1 - r_s/r) + r² (sin² d²) ---------- (15b)
  
  

  		Eq.(15b) can be used to equate dW = (r/rs - 1)-1/2dr. Integration of the equation gives W2 = 4rs(r - rs), which is a parabola function with vertex at W = 0, r = rs. Sweeping the curve around the W axis to include all values of from 0 to 2 results in a paraboloid surface as shown in Figure 09i.
  Figure 09i Worm- hole in Hyperspace [view large image]	Figure 09j Worm- hole Throat [view large image]

  Inside the wormhole, g₁₁ and g₄₄ change sign as shown in Eq.(13). The region becomes space-like, which means two events cannot be linked unless the signal propagates with greater than light speed. This peculiar property is also related to the instability of the wormhole. However it is suggested that if there is a large amount of negative mass/energy -m (in the forms of a thin spherical shell, which appears in the embedding diagram as the yellow circle as shown in Figure 09j) to sustain the structure, then creation of a wormhole may become feasible. The negative mass ensures that

  		the throat of the wormhole lies outside the horizon (since the new event horizon is now 2(M-m)G/c2), so that travelers can pass through it, while the positive surface pressure of such exotic material would prevent the wormhole from collapsing. This would allow for shortcut in space travel within the wormhole between two distant points (see Figure 09k), or for the possibility of time travel courtesy of LHC (Figure 09l).
  Figure 09k Space Travel [view large image]	Figure 09l Time Travel [view large image]

Kerr's Solution and Rotating Black Hole

---------- (15c)

Since most collapsing massive objects would have some initial angular momentum, the Kerr's metric seems to be a more realistic representation for the black hole space-time. However, it is argued that the collapsing object would lost its rotational energy in the form of gravitational wave and ends up in a perfect spheroidal shape. Whichever to be the case, followings are some comments on the Kerr's metric and its modification to the Schwarzschild's black hole:


• At r , the Kerr's metric reduces to the one for flat space-time (same as Eq.(14)):
  
  ds² = c²dt² - dr² - r² (sin² d² + d²) ---------- (15g)
  
  
• The Kerr's metric reduces to the Schwarzschild's metric in Eq.(13) naturally when a = 0, i.e., with no rotation.
  
  

It was shown theoretically that light ray is deflected by gravitational field, e.g., binding of distant star light near the sun (see Figure 09m). The angle of deflection can be computed from the equation of motion Eq.(12b), or from the equation of energy conservation: = 4GM/Rc2, where M is the mass of the source and R is the distance from the source. At the surface of the Sun, = 1.75" (derivation from Newtonian mechanics is half as much). This prediction was confirmed by Sir Arthur Eddington's 1919 solar eclipse expeditions. If the light ray comes close enough to a dense

Figure 09m Bending of Light [view large image]

object, the path will be bent so much that it runs around in a circle. For non-rotating black hole such special trajectory occurs at r = 3rs/2 = 3GM/c2. The sphere with such a radius is called the photon sphere. However, the orbit is unstable; it can be

disrupted with very small perturbation. This precarious orbit is derived from the condition d²r / d² = 0. There are two photon spheres for the rotating black hole - an outer one for light ray traveling in a direction opposite to the spin of the black, and an inner one for co-rotating light ray.


• As r descends further toward the center, the space-time metric g₄₄ (or g_tt) vanishes at
  
  r = [GM + (G²M² - a²c²cos²)^1/2] / c² ---------- (15i)
  
  

  This is called static limit. It can be intuitively characterized as the region where the rotation of the space-time is dragged along with the velocity of light. Within this region, space-time is warped in such a way that no observer can maintain him/herself in a non-rotating orbit, but is forced to become co-rotating (Figure 09n). The surface of this region is elliptical with its major axis at = /2 (the equator), and r = 2GM / c2 (= the non-

  Figure 09n Frame Dragging [view large image]

  rotating Schwarzschild's radius). The minor axis is in the directions of = 0, and (the poles), and r = [GM + (G2M2 - a2c2)1/2] / c2 (see Figure 09o).

  It is found in 2006 that a spinning superconductor generates a much larger drag than that from a regular rotating mass. The effect is attributed to some sort of "gravitomagnetism" (Figure 09n, no relation to electromagnetism) produced by massive gravitons similar to the Meissner effect produced by massive photons. The discovery may fulfill the "anti-gravity device" perpetrated by science fictions.
  
  

  	A rotating black hole is very different from a Schwarzschild black hole in that the spin of the black will cause the creation of two event horizons instead of just one (see Figure 09o). The outer event horizon occurs at r+ = [GM + (G2M2 - a2c2)1/2] / c2 ---------- (15j) The inner horizon (sometimes called the Cauchy horizon) is located at r- = [GM - (G2M2 - a2c2)1/2] / c2 ---------- (15k)
  Figure 09o Kerr's Solution [view large image]	The Ergosphere is the region between the static limit and the outer event horizon. Since this region is outside the event horizon, particles falling within the ergosphere may escape the black hole extracting its spin energy in the process.

  The separation between the two horizons is 2 (G²M² - a²c²)^1/2 / c². For a = 0, r_- =0, hence for a non-rotating black hole, the inner event horizon can be considered as fallen into the center. As the spin increases, the two horizons move toward each other and merge at r = GM / c² when a / c = GM / c². In case a / c > GM / c², there will be no event horizon, the black hole becomes a "naked singularity", i.e., it is not covered by an event horizon.

  The mere thought of such possibility makes a lot of physicists very uncomfortable. Nevertheless, the Penrose's conjecture on cosmic censorship, which forbids the occurrence of naked singularity, may not hold up any longer in view of the Kerr's solution and more recently the revelation of computer simulations (with one such results shown in Figure 09oa). Other stellar configurations that can develop into naked singularity; include inhomogeneous density (such as onionlike structure), shearing of material near a singularity, and very rapid collapsing rate. Each of such

  Figure 09oa Naked Singularity [view large image]

  case has a threshold separating the formation of black hole or naked singularity.

  Note 1: The only physical part of a black hole is the singularity. The static limit, and event horizon are not physical barrier; they only mark the imaginary boundaries between types of space.
  Note 2: A proposal in 2007 suggests that naked black hole should be detectable via the "gravitational lenses" effect. Existing telescopes should have sufficient spatial resolution to spot naked singularities in the center of the Milky Way. Other methods of detection rely on special signatures from high-energy explosions, gamma-ray bursts, and spinning rate.



• As r , a singularity develops in the form of a ring at the equator (see Figure 09p), where /2, and cos 0 in such a way that r > (a / c) cos. The Kerr's metric then becomes:
  
  ds² = (1 - 2GM / c²r) c²dt² - (2GMa / c²r) dt d - (a²/c²) (1 + 2GM / c²r) d² ---------- (15n)
  
  The severity of the singularity depends on the angular momentum per unit mass a. If a is sufficiently large it would be very mild; on the other hand it becomes a point singularity at the limit a 0. Away from the ring of singularity in the region where r ~ 0, the Kerr's metric has the form:
  
  ds² = c²dt² - cos² dr² - (a²/c²) (cos² d² + sin² d²) ---------- (15o)
  
  Interesting theoretical physics can take place around this ring singularity. Since 1 g₁₁ = cos² in Eq.(15o), the spatial curvature is negative, which acts like a repulsive force. One consequence is that nothing can actually fall into this region unless approaching in a trajectroy along the ring's side. Any other angle and the negative spatial curvature actually produces an antigravity field that repels matter. It could be the mechanism that produces the jets observed in many black holes.




The Penrose diagram in Figure 09p summarizes all regions of space-time associated with a rotating black hole. A penrose diagram is not meant to accurately portray distances, it describes only the causal structure. Only radial and time directions are represented while angular directions are suppressed. The units of space and time are scaled in such a way that any object moving at the speed of light will follow a path at an angle of 45º to the vertical. All possible paths for physical objects must stay closer than 45º to the vertical. The green diamond represents our entire Universe over its entire history and destiny, from the infinite past to the infinite future (it takes an infinite amount of time to cross the outer event horizon from the view point of a distant observer). The purple diamond is the region between the outer and inner event horizons, where everything plunges inward. The red area denotes the space between the inner event horizon and the ring singularity, where the space-time re-acquires its normal characteristic with rising curvature toward the ring singularity. The other half in yellow is the region

Figure 09p Penrose Diagram [view large image]

inside the ring singularity, where the gravity becomes repulsive. The blue diamonds can be interpreted as another universes or another part in our own universe. This concept is similar to the wormhole in the Schwarzchild's solution.

As depicted in the Penrose diagram, the space-time metric g11 (or grr) changes sign when crossing over the first event horizon, and then reverses back again at crossing over the second horizon. Thus between the two horizons, space and time exchange places. Instead of time always moving inexorably onward, the radial dimension of space moves inexorably inward to the Cauchy horizon. After that the Kerr solution predicts a second reversal, which implies no more plunging inward. In this strange region inside the Cauchy horizon the observer can, by selecting a particular orbit around the ring singularity, travel backwards in time and meet himself, in violation of the principle of causality (cause must precede effect). It is surmised that a closed time-like curve (CTC, Figure 09q) to loop back to the past is possible in a heavily curved space-time (Line A in Figure 09p). Another possibility admitted by the equations for the observer in the central region is to plunge through the hole in the ring into an antigravity region (Line B). Or he can travel through two further

Figure 09q Closed Time-like Curve (CTC)

horizons (or more properly anti-horizons), to emerge at coordinate time t = – into some other universe (Line C). These exotic properties of rotating black holes have inspired several science fiction stories.





A more realistic rotating black hole would have an accretion disk around the equator and a pair of jets ejecting along the poles. An accretion disk is matter that is drawn to the black hole. As matter is gradually pulled into the center, it gains speed and energy. It can be heated to temperatures as high as 3 billion K by internal friction, and emit energetic radiation such as gamma rays. Some of the particles that has funneled into the disk-shaped torus by the hole's spin and magnetic fields, can escape the black hole in the form of high speed jets along the poles (Figure 09ra).

Figure 09ra Disk and Jets [view large image]

Black Holes are very difficult to detect because they are black. It would be a definitive proof of their existence if we can see the "shadow" in the form of a dim spot in the centre of the glowing accretion disc (Diagram c in Figure 09rb). Most of the circumstantial evidences come from observing high velocity objects moving tightly in small space as shown in Diagram a, Figure 09rb. Other evidences involve detecting gravity waves from merging black hole(s) or the double image from a spinning black hole (see Diagram b and d in Figure 09rb), which throws out huge jets of debris to advertise its presence. The final answer to whether our black hole ideas are correct would be sending a probe to a nearby candidate and have it transmit data as it makes its final plunge.

Figure 09rb Black Hole Detections [view large image]

A Scenario for Time Travel

Recently in 2006, a scenario for time travel has been proposed without relying on rotating black holes or exotic wormhole tunnels. The idea is based heavily on the Superstring theory, according to some versions of which, our universe is a four-dimensional membrane or "brane" embedded in a higher-dimensional hyperspace called the bulk. Almost all matter and force-carrying particles are trapped on the 3-D brane, where they are contrained to travel at the speed of light or lower. However, sterile neutrinos and gravitons are particles that can access the hidden dimensions and travel faster than light (Figure 09s). From some view points or frames of reference, this is equivalent to time travel (see Figure 04c, event 4).

Figure 09s Time Travel [view large image]

The main ingredients of this theory are summarized in the followings:

For many years, physicists believed that extra dimensions were acceptable, but only if they were finite in size, either curled up or bounded between branes. An infinite extra dimension was supposed to destablilize everything around us. It is now known that infinite extra dimension from a single brane (like our 3D brane) is possible if that extra dimension is heavily warped so that the gravitons are localized near the brane in a small region no more than the Planck length scale of 10-33 cm (see Figure 09t). Thus, instead of the requirement for a heavily curved space-time in our 3D universe, or a folding one such as shown in Figure 09k, a model can be conceived such that the extra dimension in the bulk is seriously warped. It is found that the extra dimensions can be distorted by exotic matter in such a way that anything moving through the bulk can travel faster than the speed of light. Such off-brane short cuts can appear as "closed time-like curves" (CTC) as shown in Figure 09q.

Figure 09t Extra Dimension [view large image]

• This has dramatic consequences for inhabitants stuck on the 3D universe. To them, any entity that takes a short cut through the bulk appears to vanish and then pops up again at some point far sooner than it could have had it kept to the 3D universe, and in some other frames of reference, it has travelled backward in time (see Figure 04c, event 4).
• However, according to the Superstring theory, only closed strings such as the sterile neutrinos and gravitons can wander into the bulk. We may not be able to do time travel ourselves, but we can study time travel experimentally by manipulating these particles. The problem is that no one has ever spotted a graviton or a sterile neutrino.
• Nevertheless, experiment to test time travel has been proposed via the mechanism of neutrino oscillation, which could change ordinary neutrinos into sterile neutrinos and vice versa. The odds of this happening increase whenever the density of the material the neutrinos are travelling through changes abruptly. Thus, it is suggested that a beam of ordinary neutrinos would be sent through the Earth, from South Pole to a detector located at the equator. Some of them flipped into sterile neutrinos going through the bulk and emerge as ordinary neturino (via another flipping), which may appear as moving at greater than light speed or arrived before setting off.
• Critics point out that even if future technology can manipulate sterile neutrino in the next 50 years or so, the problem with the hypothetical exotic matter (to produce the warped space-time) is only shifted from our universe to unknown dimensions. Concealing the problem in the bulk is little better than a scenario in which it is out in the open. However, if the experiment is successful, it will verify many conjectures such as higher dimensions, some versions of the Superstring theory, sterile neutrinos, exotic matter in higher dimensions, and time travel albeit only for sterile neutrinos and gravitons.

Hawking Radiation

According to Stephen Hawking himself, an inspiration came to him before going to bed one evening in 1970 (getting into bed is a rather slow process with his disability). He suddenly realized that since nothing can escape from a black hole, the area of the event horizon might stay the same or increase with time but it could never decrease. In fact, the area would increase whenever matter or radiation fell into the black hole. This non-decreasing behavior of a black hole's area was very reminiscent to that of entropy, which measures the degree of disorder in a system. One can create order out of disorder, but that requires expenditure of effort or energy such that there is an overall increase in disorder. In simple mathematical terms these statements can be expressed in differential forms as outlined below:


1. From the definition of the Schwarzschild radius R = 2Gm/c², and its surface area A = 4R², we can derive an expression for the small change of the area dA by throwing in a small amount of mass dm:
   dA = (32G²/c⁴)mdm ---------- (16a).
2. From E = mc², Eq.(16a) can be rewritten to:
   dE = (c⁴/32mG²)dA.
3. By definition, the corresponding increase in entropy is:
   dS = dE/T = (c⁴/32mTG²)dA ---------- (16b),
   where T is the temperature of the black hole in ^oK. But it has been shown in the topic of Black Hole Entropy that in term of Planck area:
   dS k_B(c³/G)(dA/4) ---------- (16c).
4. Equating Eqs.(16b) and (16c), we obtain a formula relating the mass m and temperature T for a black hole:
   T = hc³/(16²Gk_Bm) ---------- (16d).

		horizon may allow one member of the virtual particle / anti-particle pair to fall inside with negative energy; while the other escapes as a real particle with a positive energy according to the law of energy conservation. This is known as Hawking radiation (see Figure 09u); it is the first successful attempt to combine general relativity and quantum theory. The flow of negative energy (or mass) into the black hole would reduce its mass. As
Figure 09u Hawking Radiation [view large image]	Figure 09v Black Hole Evaporation [view large image]	the black hole loses mass, the area of its event horizon gets smaller, but this decrease in the entropy of the black hole is more than compensated for by the entropy of the emitted radiation, so that the second law of thermo-dynamics is never violated.

This phenomenon of Hawking radiation also occurs in the event horizon created by an accelerating observer. Figure 09w shows that light ray emitted at certain distance can never catch up with the observer and thus an event horizon exists beyond which the observer cannot communicate. Theoretical arguement suggests that even in empty space, the observer will be able to detect radiation from the event horizon. A simple formula is derived to express the relationship between the acceleration a and the temperature T:

Figure 09w Event Horizon of an Accelerating Observer [view large image]

T = a (/2kBc). It is suggested that members of the correlated virtual photon pairs are separated by the event horizon. As a result part of the information is missing, the observer detects random motion associated with the temperature. In this case the energy is extracted from the acceleration, which according to general relativity, is equivalent to gravitation.

Black Hole and Information

• According to general relativity anything that crosses the event horizon is trapped inside forever lost to the outside world.
• In spite of his own discovery of Hawking radiation, which can return matter-energy back to the outside world, Hawking himself insists that information would be destroyed by the black hole since the Hawking radiation does not convey any information about the interior.

Critics argue that according to the rule of conservation of information (which is related to the time reversal symmetry in both classical and quantum mechanical laws in physics), information cannot be destroyed although it may be scrambled (Figure 09xa). But Hawking was sufficiently confident about his position to place a bet with theoretical physicist John Preskill on the fate of information. Since then it has been shown that microscopic quantum ripples at the event horizon can encode information inside the black hole, so there is no mysterious information loss as the black hole evaporates. In 2004, Hawking called a press conference and announced to the world that he had changed his mind. Black holes did not, after all, irreversibly annihilate information. The bet with Preskill was duly settled in the form, appropriately enough, of an encyclopedia.

Figure 09xa Black Hole Information [view large image]

• In late 2006, 't Hooft and Susskind proposed the "Principle of Black Hole Complementarity" in which both sides are correct. For the outside observer, matter (the elephant in Figure 09xb) would be gradually reduced to thermal radiation

	(the Hawking radiation) at the event horizon (as it takes an infinite time for the elephant to cross such boundary according to observer a) and returns as scrambled information. While for observer b, the elephant crosses the event horizon into the black
Figure 09xb Black Hole Complementarity [view large image]	hole, nothing untoward happens until the tidal force takes over ... information is forever lost.

Meanwhile, other research in superstring theory finds the black hole to be a "fuzzball" (Figure 09y). The modified black hole does not possess a sharp event horizon; information can be stored in the strings and imprinted on outgoing Hawking radiation. Models of black holes from superstring theory also cast doubt on the idea of the singularity (at the center of the black hole). Yet another scheme suggests that information might leak out by means of quantum teleportation between the entangled pair of virtual particles (one of which has escaped while the other is trapped inside the black hole). However in the theory of loop quantum gravity, it has been shown that the information trapped in a black hole will be unable to escape via Hawking radiation. But it will survive, eventually rejoining the rest of the universe when the black hole evaporates. So it looks unlikely that Hawking will be able to recover his bet.

Figure 09y Fuzzball [view large image]

It seems that Stephen has been a compulsory gambler. Another physicist had made a similar wager with him in 1980 about the fate of information falling into a black hole. Figure 09z is a copy of the contract together with the eventual concession by him completed with his signature plus a fingerprint. The document is couched in highly technical terms such as S-matrix and $-matrix. In simple language S-matrix is a mathematical entity that allows a process to run backward, thus retrieving the original information. The $-matrix ($ denotes "NOT" S) is something invented by Stephen to overturn such rule for the case when the process involves a black hole.

Figure 09z

Stephen's Wager [view large image]

Standard Cosmology

• The space-time interval is associated with the Robertson-Walker metric. It has three forms depending on the curvature of the 3 dimensional space:
  
  ds² = c²dt² - R(t)² (dr² + w² (d² + sin² d²)) ---------- (17)
  
  where w = sin(k^1/2 r) / k^1/2, k (in unit of cm^-2) can be considered as the total energy of the universe, and R(t) is called the scale factor. It expands the coordinate grid as a whole with the expansion of the universe. Such grid is referred to as comoving coordinates. Thus if we denote dr to be the comoving distance between two points, then dx = R(t)dr would be the physical distance actually measured by a ruler.
  
  For flat space with zero curvature, k = 0, and w = r, where r ranges from 0 to infinity.
  For closed space with positive curvature, k > 0, w = sin(r), where r ranges from 0 to 2.
  For open space with negative curvature, k < 0, w = sinh(r), where r ranges from 0 to infinity.



• In term of the Robertson-Walker metric and with the energy-momentum tensor T₀₀ = , the gravitational field equation can be written in a form similar to the energy equation (K.E.+P.E.=E):
  
  (dR/dt)² - 8GR² / 3 = - kc² ---------- (18a)
  
  which can be re-arranged into the form:
  
  k² c² = H² R² ( - 1) ---------- (18b)
  
  where H(t) = (dR/dt) / R is the Hubble's parameter, = / _c, and _c = (3 H₀²) / (8G) is the critical density, which corresponds to the total energy density for a flat universe. Eq.(18b) shows that
  
  for flat space, = _c, = 1, and k = 0;
  for closed space, > _c, > 1, and k > 0;
  for open space, < _c, < 1, and k < 0.


While the Hubble's parameter H(t) depends on the time t, its value at the current epoch t₀ is often denoted as H₀, and called the Hubble's constant; its inverse T = 1/H₀ is called Hubble's time - the time since the Big Bang without retardation or acceleration. The scale factor for the current epoch is often taken to be unity for convenience, i.e., R₀ = 1 by equating the radius of the universe r₀ = (9GM/2)^1/3 t₀^2/3 = 10²⁸cm for t₀ = 13.7 billion years according to Eq.(19a).


• More generally, when several types of matter coexist in the universe, the following consistency relation must be satisfied:
  _jj + _k = 1
  where the sum is over the various types of matter and _k = - kc²/(R₀H₀)². Present cosmological observations yield:
  
  _b ~ 0.04 for brayons (ordinary visible and nonluminous matter),
  _d ~ 0.26 for exotic dark matter,
  ~ 0.7 for dark energy,
  ~ 5 x 10^-5 for photons (radiation), and
  _k = 0.



• For a homogeneous and isotropic universe = M / (4R³r³/3) ---------- (18c)
  
  where M may be interpreted as the total mass of the universe, Eq.(18a) becomes:
  
  (dR/dt)² - 2GM / Rr³ = -kc² ---------- (18d)
  
  The solution for this equation is:
  
  for k = 0,     R = (9GM/2r³)^1/3 t^2/3,      t = 2/3H ---------- (19a)
  
  for k > 0,     R = R_o (1 - cos(k^½µ)),     ct = R_o (µ - k^-½sin(k^½µ)) ---------- (19b)
  where R_o = GM / |k|c²r³, and µ is defined by c dt = R(µ) dµ.
  
  for k < 0,     R = R_o (cosh(|k|^½µ) - 1),     ct = R_o (|k|^-½sinh(|k|^½µ) - µ) ---------- (19c)


For the radiation (relativistic matter, i.e., particles with kinetic energy higher than the rest mass energy) dominated universe, the radiation pressure p = /3. Starting from the thermodynamic relation dU = -pdV, where U = V, and V is the volume of the system (the universe in this case), it can be shown that 1/R⁴, Eq.(18a) (for the case k = 0) becomes dR/dt 1/R, the solution for which is R t^½. Incidentally, since the temperature T is related to the energy density as T⁴, thus T 1/R.


• The red shift caused by the cosmic expansion can be expression in terms of the scale factor R [see Figure 10a, which shows the red shift of the Balmer series of the hydrogen line spectra (dark lines, denoted by H, ...) superimposed on the continuous spectra (colored), the arrows indicate the direction of increasing values for the corresponding variables]:
  
  z = ₁ / ₂ - 1 = R(t₂) / R(t₁) - 1 ---------- (19d)
  
  

  where t2 denotes the current epoch, 2 is the original wavelength (as measured on Earth), 1 is the wavelength of a spectral line that was emitted at time t1 (red shifted) in the distant past, R(t2) is often taken to be 1 for convenience, then R(t1) is less than 1 in the past and is often just denoted as R(t). The quantity (z + 1) can be considered as the amount of stretching during the intervening time for the light to travel from there to here. It is a curious relativistic effect that the velocity and distance remain finite approaching the velocity of light and the event horizon respectively even though

  Figure 10a Cosmic Red Shift [view large image]

  the amount of stretching becomes infinity. The relativistic expression for z+1 (In terms of the recession velocity v) is: z + 1 = [(1+ v/c) / (1- v/c)]1/2.

  From the observed value of the Hubble constant Ho = 71 km/sec-Mpc for the current epoch, and by substituting the speed of light c for v in the Hubble law: v = Ho d, it follows that the rate of cosmic expansion would be exceeding the speed of light beyond a distance DH = c/H0 = 1028cm. However, there is no contradiction, because special relativity states that no signal can propagate faster than the speed of light, but the expansion of the universe cannot be used to propagate a signal although it can move with superluminal speed. We will not be able to see this region anyway because the event horizon is located right at the boundary (Figure 10b), if we take the Hubble time T = 1/H0 = 13.7 billion years (from WMAP modeling) to be the age of the observable universe.

  Figure 10b Event Horizon [view large image]

• There are various kinds of cosmic distance according to the way it is measured (see Distance Measures in Cosmology by David W. Hogg for a review in details). They have different dependence on the redshift z, and yield different limiting values as z . Table 02 below is a summary of all these distances. The formulas in terms of z are derived from the "matter only" flat-space model, which can be expressed in close (analytic) form. The more realistic solution including dark energy has to be evaluated numerically as shown in Figures 10c. Figure 10d illustrates schematically the

  			different kinds of distance. Figure 10e depicts the lookback dis-tance and time etc pictorially (not to scale). The Hubble distance in Table 02 is defined as DH = cT = 13.7x109ly
  Figure 10c Types of Cosmic Distance [view large image]	Figure 10d Cosmic Distances	Figure 10e Lookback Distance and Lookback Time [view large image]

  The comoving distance is the fundamental distance measure in cosmology since many others can be derived in terms of it. Within the framework of general relativity, it can be expressed as:
  D_C = D_H dz' / [_M(1+z')³ + _k(1+z')² + ]^1/2
  where _M = 8G₀/3 H₀², _k = -kc²/ H₀², and = c²/3 H₀².
  Thus, a theoretical comoving distance can be computed as a function of the redshift z from the density parameters 's resulting in many cosmological models with different combinations of these parameters. Similarly, the lookback or light travel distance is given by:
  D_T = D_H dz' / (1+z')[_M(1+z')³ + _k(1+z')² + ]^1/2
  The difference between the two definitions is that the former is derived from the conformal time , while the latter is related to the regular time t. The conformal time is introduced because of a peculiar feature in cosmic expansion. It has been mentioned that the physical distance dx is related to the comoving distance dr by the formula dx = R(t)dr. It follows that the velocity dr/dt = (1/R)dx/dt, then for dx/dt = c, dr/dt > c in case R < 1. The conformal time defined by
  dt = Rd ---------- (19e)
  is invented to address this problem of exceeding the velocity of light. Since by using the conformal time dr/d = dx/dt, it would not have the problem when dx/dt = c. The integration of Eq.(19e) then leads to:
  cd = cdt/R(t) = D_C
  while cdt = D_T
  
  

  Type of Distance	Definition	Function of z (all matter)	Distance as z
  Comoving or Conformal (DC)	Distance scale expanding with the universe at the current epoch	2[1-1/(1+z)1/2] DH	2DH = 27.4x109ly (DC = 47.2x109ly for more realistic model)
  Coordinate or Proper or Physical (DP)	Distance scale expanding with the universe at redshift z	DC/(1+z)	0
  Angular Diameter (DA)	Intrinsic linear size L / angular size	DC/(1+z)	0
  Luminosity (DL)	Calculated from apparent brightness	(1+z) DC
  Light Travel or Lookback (DT)	Light travel distance from source to here	(2/3)[1-1/(1+z)3/2] DH	(2/3)DH = 9.1x109ly (DT = 13.7x109ly for more realistic model)
  Light Travel Time or Lookback Time (tl)	Time difference between source & here	(2/3)[1-1/(1+z)3/2] T	(2/3)T = 9.1x109yrs (T = 13.7x109yrs for more realistic model)

  Table 02 Types of Cosmic Distance

  	The diagram on the left of Figure 10f shows the Type Ia supernova raw data up to a redshift of about 1.5 (or down to 0.4 in term of the scale factor). After various adjustments for these observations, the data points agree remarkably with a theoretical curve as shown in the diagram on the right. This model is computed with matter-energy composition and spatial curvature estimated by WMAP.
  Figure 10f Type Ia SN Data [view large image]	Note that (by flipping the curve to the left) Figure 10f represents a short segment of one of the cosmic models in Figure 10g.

  The various data reductions include:
  
  1. Change redshift to scale factor.
  2. Correction for over estimate of luminosity distance by expansion-induced dimming of light.
  3. Conversion from coordinate distance to light travel time or lookback time.
  4. Average over multiple galaxies.
  
  

  Figure 10g shows the three types of universe - open, flat, and closed. Universes that are too far above the critical divide expand too fast for matter to condense into stars and galaxies; such universes therefore remain devoid of life. Those that fall too far below the critical divide collapse before stars have a chance to form. The shaded area indicates the range of cosmological expansions and epochs in which observers could evolve. By measuring the sum of the angles in a huge narrow triangle with one vertex on Earth and the other two at the points of emission of CMBR (see insert in Figure 10g), the WMAP data yields a value of 180o indicating that the universe is

  Figure 10g Types of Universe [view large image]

  flat. Such conclusion is further supported by the acceleration of Type Ia supernovae, which imply a precise amount of dark energy needed to make the universe flat (with the total matter-energy density equals to the critical density).

  Note: For closed space the sum of the angles in a triangle would be greater than 180^o; it is smaller than 180^o for open space; it equals to 180^o exactly for flat space.

Cosmological Constant and de Sitter Universe

(dR/dt)2 + kc2 = 8GR2 / 3 + R2 c2 / 3 ---------- (20a) where is the cosmological constant. It can be expressed in term of the corresponding density : = 8G / c2. Assuming a flat universe, current observations of distance supernovae, the cosmic microwave background radiation, and the dynamics of galaxies together favor a value of = 0.7 c, the numeric value for is about 1.3x10-56 cm-2. Past attempts to identify the cosmological constant with the vacuum energy of the various quantum fields was not very successful.

Figure 10h Cosmological Constant [view large image]

The effect of the cosmological constant on the cosmic expansion is summarized in Figure 10h.

Ht

where H = (/3)1/2c = (dR/dt)/R is the Hubble constant, and R0 is a constant.. This solution shows the weird property of a constant matter-energy density , i.e., vacuum energy is continuously being created to fill up the void in the wake of the expansion. The idea is similar to the steady state universe¶, which requires the continuous creation of matter. Since the event horizon is dh = c / H = 1 / (/3)1/2 = constant; therefore, like the earth's horizon, the de Sitter horizon can never be reached - it is always a finite distance away. The de Sitter universe with k = 0 starts at t - from a singularity. It reaches a size of Ro at t = 0, and expands to infinity as t +. Alternatively, Ro at t = 0 can be taken as the initial condition. The universe then can either grow or decay exponentially. Note that Eq.(20b) is time reversal invariant. For some reason, this universe chose to grow exponentially in the positive time direction.

Figure 10i de Sitter Universe with k = +1 [view large image]

For k = +1, R(t) = (c/H) cosh(Ht) (c/2H) eH|t| as t .

Theory of Cosmic Inflation and Acceleration

Astronomical observations over the last 20 years have indicated that the universe has experienced two periods of acceleration (see Figure 10j). The first one was a very rapid expansion soon after the Big Bang. It is usually referred to as inflation. The dark energy accelerated expansion is gentler, and has occurred only recently in the current epoch. The inflation is explained by an as yet un-identified scalar field (inflaton) undergoing a phase transition. While there are three categories of theory for the accelerated expansion: modifications to general relativity perhaps with extra dimensions, a cosmological constant, and a universal evolving scalar field. None of these offers a satisfactory explanation.

Figure 10j Cosmic Inflation [view large image]

The followings provide a mathematical description of cosmic inflation and expansion without invoking a detailed mechanism.

• If we assume a homogeneous and isotropic universe, with the total mass to be constant as expressed in Eq.(18c), then Eq.(18a) can be recast into a form that shows the acceleration / deceleration explicitly:
  
  (d²R/dt²) / R = - 4G / 3 ---------- (20c)
  
  This formula shows that the matter dominated universe in general would experience deceleration as the term on the right is always negative.



• In the presence of other forms of matter-energy, a pressure term p has to be added into Eq.(20c):
  
  (d²R/dt²) / R = - (4G / 3c²) (c² + 3p) --------- (20d)



• The matter-energy density and the pressure p are usually related by the equation of state, which can be expressed as p = wc², thus Eq.(20d) can be rewritten as:
  
  (d²R/dt²) / R = - (4G / 3) (1 + 3w) ---------- (20e)
  
  

  For relativistic matter, w = 1/3. For non-relativistic matter, w = 0. For the de Sitter universe, w = -1. The expansion of the universe will accelerate if w < -1/3, when the right-hand side in Eq.(20e) becomes positive. The negative pressure is a characteristic of expansion under constant density as shown by a simple example. It is also known as false vacuum. In the simplest version of the inflationary paradigm a single scalar field (inflaton) dominates the energy density of the universe. To achieve the acceleration condition w < -1/3 and the observed properties of our universe, the inflation must evolve slowly such that the potential energy dominates over the kinetic energy for a sufficient part of the inflation. Figure 10k shows a theory (the old one) that doesn't work because the scalar field (the Higgs field) evolves rapidly.

  Figure 10k Theories of Inflation [view large image]

  While the newer theory is just right creating an universe as we see it today. This scenario remains ambiguous as the precise form of the scalar field is unknown, and there is still nagging doubt about the occurrence of inflation.

  
  
  
• Another way to achieve acceleration can be derived from the cosmological constant as shown in Eq.(20a). Assuming constant mass (including both baryonic and dark matter) within the expanding universe, Eq.(20a) can be simplified to (for k=0) :
  
  dR/dt = R(_m/R³ + )^1/2 ---------- (20f),
  
  which can be recast to a form expressed the cosmic acceleration explicitively:
  
  (d²R/dt²) = R[- _m/(2R³) + ]
  where the time is in unit of 1/H₀=13.7x10⁹ years. This is a very simple second order differential equation suitable for numerical calculation on a home computer using the Basic computer language. For example, the effect of the density parameters 's on the age of the universe can be estimated by running the computer program with different combinations of the parameters. In particular without solving the equation numerically, just by equating this equation to zero, i.e., at the moment when deceleration reverted to acceleration, yields the change over epoch at z = (1-R)/R = 0.75 (or about 8 billion years after the Big Bang) for _m = 0.27, and = 0.73.
  
  Effort to identify the cosmological constant with the vacuum energy of the various quantum fields is not very successful. The simplest versions of quantum theory predict far too much energy - 10¹²⁰ higher than the observed value by one estimate.



• The explanation of the current acceleration in term of an evolving scalar field presents a tremendous hierarchy problem. The very high energy (10¹⁶ Gev) associated with the scalar field in the inflation period requires an unreasonable amount of fine-tuning to account for the extraordinarily low energy (10^-48 Gev) of the scalar field in the current epoch.



• A different explanation was proposed by a research team in 2005, they links the dark energy acceleration to the gigantic ripples in space-time created in the epoch of inflation. In effect, it is like hobbling amid a huge undulating wave in the ocean, you don't actually see the wave (the wavelength is too long) - all you feel is the swell moving up and down.

Static Universe

community had recognized the expansion of the universe by late 1920s. He can be excused for making this mistake because astronomical observation at earlier time indicated a homo-geneous distribution of objects in the sky and the view did not seem to change over "long" time. It induced him to assume a constant density in Eq.(18a). Simple mathematics as well as from solution of the equation§ shows that such model describes an expanding universe. In order to avoid this "unwanted" predicament, he introduced a counter term to make the right-hand side identically equal to 0 by fudging the cosmological constant with = - 8G/c2 (~ negative pressure) in Eq.(20a). Then dR/dt 0, and R = constant for k = 0. With the discovery of cosmic acceleration, it is fashionable again to re-introduce the cosmological

Figure 10l Einstein and His Cosmic Blunder [view large image]

constant into the model universe. The crucial difference is that we now know the density is not a constant; it varies with time as the universe expands. The two terms on the right-hand side of Eq.(20f) equal to each other only momentarily about 8 billion years after the Big Bang.

Angular-Size Redshift Relation

The angular-size redshift relation describes the relationship between the angular size observed on the sky of an object of given physical size, and the object's redshift. In elementary Euclidean geometry the relation between angular size , linear size l and distance d (close to the Earth) would simply be given by the equation: = l / d In an expanding universe the distance d is a function of the redshift z: d = (c/Ho) {qoz + (qo - 1)[(1 + 2qoz)½ - 1]} / [qo2(1 + z)2]

Figure 10m Angular-Size / Redshift [view large image]

where qo = (4G / 3) / H02 is the deceleration parameter with the substitution by Eq.(20e). In standard cosmology qo = 0.5 for flat space, qo > 0.5 for closed space, and qo < 0.5 for open space (see Figure 10m, angular size in unit of mas = milliarcseconds).

These limiting cases clearly demonstrate the curious effect that the angular size of an object becomes larger as it is further away from Earth. It appears to decrease with distance only for nearby objects. Figure 10n shows the infrared blobs produced by the first stars (high z objects). It is suggested that the appearance of the puffy blobs with large angular size is caused by the expansion of the universe with z > 1.6 as shown in Figures 10k and 10m.

Figure 10n Infrared Blobs [view large image]

In principle, the angular-size redshift relation can be used to select the type of space for our universe. However, it is notoriously difficult to collect reliable data in practice because the astronomical yardstick can vary in size and in luminosity over time (the evolutionary and selection effects). In addition, we can only measure the projections on the celestial surface according to the orientation of the objects. All past attempts using data from galaxies, the separation of the lobes of radio sources, quasars, and radio galaxies produced inconclusive results. The observational data in Figure 10o are based on selective compact radio sources. The best fitting regression analysis gives a value of qo 0.21. More recent study in 2004

Figure 10o Model with Cosmological Constant [view large image]

with ultra-compact radio sources find close match for the model of an universe with cosmological constant, m = 0.24, = 0.76, and spatially flat. The observational data range between 0.6 < z < 2.7 with a population mean for the linear size l ~ 6.2 pc. It indicates a "switch over" from deceleration to acceleration at z = 0.85 (see Figure 10o).

Euclidean Space

In 3-D Euclidean space, the formula for a sphere of radius a located at (0,0,0) has the form: x2 + y2 + z2 = a2 with the metric ds2 = dx2 + dy2 + dz2 = a2 (d2 + sin2 d2) ---------- (20g) in spherical coordinates (r, , ), where the radius r = a is constant (Figure 10p). The sphere has only two degrees of freedom. Further simplification is possible by considering only those circles around the z-axis, or z = a cos = constant, which implies = constant. In this case

Figure 10p 2-D Sphere [large image]

ds2 = a2 sin2 d2. The length of the circumference is 2 a sin, which varies from 0 to a maximum of 2a and falling back to 0 as varies from 0 to /2 and then to . The point at = 0 or (sin = 0 at the North and South poles) is not a singularity as it can always be removed by re-defining the origin of reference.

The no boundary proposal makes use of Eq.(20i) and suggests that at the beginning the universe was running with as the time component from = 0 to /2 in the Euclidean space. At /2 the character of the time component underwent a transformation, e.g., /2 + ict/a. Since then the 4-D hypersphere has changed into a Minkowski hyperboloid described by the metric: ds2 = -c2dt2 + a2 [cosh2(ct/a) d32] where the 3-D hypersphere is expanding as "a cosh(ct/a)", and approaches a de Sitter universe in the form (a/2)ect/a as ct/a +. Such a hybrid universe is shown in Figure 10q. Thus, there is no singularity at the beginning of time. That particular region becomes as smooth as the North

Figure 10q Hybrid Uni-verse [view large image]

(South) pole on Earth. It is suggested that the (red) region in Euclidean time represents the moment of nucleation via tunneling through some sort of energy barrier from nothing (in the form of vacuum fluctuation for example).

Five Dimensional Space-time

It is found that when 4-dimensional theory is formulated in a 5-dimensional space-time, novel features emerge as the theory is reduced back to 4-D. The Kaluza-Klein theory introduced in the 1920s is the most famous example, which unifies the theories of gravitation and electromagnetism. The technique of compactification to hide the extra-dimension is now used extensively in the superstring theory. It is from the process of compactification that produces various particle properties.

Figure 10r 5-D Space-time [view large image]

If experimenters discover new heavy particles with the same charges as familiar ones and masses that are similar to one another, those particles will be strong evidence of extra dimensions. If such particles also occur at regular intervals of mass, it would very likely mean that a simple curled-up dimension has been discovered. But more complicated extra-dimensional geometries will yield more complicated patterns of masses. If enough such particles are discovered, the KK particles would then reveal not only the existence of extra dimensions, but also the their sizes and shapes. Since no KK particle has been detected so far, it seems to indicate that the size of the curled-up dimension could be very small.

Figure 10s KK Tower [view large image]

In unit of ev, the mass of the KK tower Mn = (n/R) x 1.2 x 10-4 ev. Thus, the new LHC collider (to be operational in 2007) with available energy up to 14 Tev can probe the curled-up dimension down to the size of about 10-17 cm.

Several conclusions can be drawn from this simple example:


• The compactification of a dimension creates a quantization of the momentum corresponding to the compactified coordinate. The momentum is labeled by integers.
• The mass spectrum in the space-time dimensions that are not compactified is shifted by an effective mass term coming from the compactified dimension.
• The radius of the compactified dimension can be totally arbitrary as long as it can hide that dimension from observation.
• In general the components with from 1 to 4 will recover the original theory with additional term. The = 0 component will generate some kind of physical property. In some other theories when there are cross terms between the = 1, ...,4 and the = 0 components, other theories will emerge from the reduction.

Some comments on the Kaluza-Klein theory:


• The extra dimension is not curl-up and satifies the periodic condition. It is proposed instead that g_ik should not depend on x₀, which implies g₀₀ is a constant.
• The reduction of the (, ) (from 1 to 4), (0, ), and (0, 0) components of the curvature tensor respectively generates the gravitational, electromagnetic, and scalar fields.
• The reduced equations for gravity is similar to the original gravitational field equations in Eq.(12a). The additional term is the electromagnetic stress-energy tensor, and k ~ g₀₀ is the gravitational constant.
• The electromagnetic equations are modified by a factor of (-g)^1/2, where g is the determinant formed from g_ik.
  F = g_0, - g_0, is the electromagnetic anti-symmetric tensor defined in the original equations. Indices separated by a comma will denote differentiation with respect to the corresponding coordinate.
• The massless scalar field is called radion or dilaton. It can be interpreted as the length or size of the fifth dimension as a function of the usual four dimensions of space-time. In string theory, the dilaton always appears together with gravity.
• The equation of motion in the reduced 4-D space-time takes the form:
  
  
  
  where ₀ is the mass density. The interaction terms involve the gravitational field interacting with the energy-momentum density, and the electromagnetic field interacting with the charge-current density.

Nowadays, we know that electromagnetism and gravity are far from being the whole story. A satisfactory unified theory must accommodate a good deal more like the weak and strong interactions. In fact, five dimensions are not enough; we might just manage with ten (such as in the string theory). The Kaluza-Klein method has been generalized to Yang-Mills theory in (4+N)-dimensional spac-time. If supersymmetry is added to the formulation, the new components of the super curvature tensor emerge as

Figure 10t Supergravity [view large image]

quarks and leptons. Such formulism is called supergravity. The decomposition is illustrated in a much simplified form in Figure 10t.

An interesting application of warped 5th dimension has been developed by Lisa Randall. In this model, the 5th dimension is located in between two 3-D branes. It is found that the extra dimension is severely warped in the form of anti de Sitter space with positive curvature by the presence of positive energy Gravitybrane and negative energy Weakbrane even though the branes themselves are completely flat (see Figure 10u). The strength of gravity depends on the position of the 5th dimension. As shown in Figure 10u (in term of graviton's probability function), it can be very strong on the Gravitybrane but becomes feeble on the Weakbrane where all the forces and particles in the Standard Model are confined. Only the gravitons can move anywhere in the branes and in the bulk. This model

Figure 10u Warped 5-D Space-time [view large image]

explains why gravity is weak in our world although it can be very strong in another brane. As the number of gravitons decreases exponentially, the separation between the two branes in the order of a few Planck length is sufficient to explain the hierarchy problem of

Gravitational Wave

dimensional space-time. This equation looks similar to the electromagnetic wave equation except that it is now a second rank tensor field (with 10 components) instead of the more familiar vector field. It is responsible for many different characteristics in these two kinds of field. Figure 10v shows the differences in polarization and radiation pattern. There are two polarization states in gravitational wave. They alternatively squeeze and stretch the interacting particles shown as white circle in the diagrams (with direction of propagation perpendicular to the viewing page). Table 03 compares the properties of these two kinds of wave.

Figure 10v EW and GW [view large image]

Table 03 Electromagnetic and Gravitational Waves

Gravitational wave have never been observed because of low radiation power and weak interaction strength. A rod about 1 meter long spun at the verge of breaking would radiate perhaps 10-30 erg/sec. The cross section for the interaction between gravitational wave of ~ 104 cycles/sec and an ammonia molecule is roughly 10-60 cm2. Figure 10w is the schematics of a gravitational wave bar detector. The impinging gravitational wave excites the fundamental longitudinal resonance (at ~ 1000 Hz) of the bar, kept at low temperatures. The induced vibration of the bar end face is amplified mechanically by the

Figure 10w GW Detector [view large image]

resonant transducer, which also converts the signal into an electromagnetic one. The signal is then amplified and acquired (see Figure 10w). It is suggested that large-scale astronomical motions of matter could generate appreciable gravitational energy flux.

	The binary pulsar PSR1913+16 was discovered in 1975. This system consists of two compact neutron stars orbiting each other with a maximum separation of only one solar radius. The rapid motion means that the orbital period of this system should decrease on a much shorter time scale because of the emission of a strong gravitational wave. The change predicted by general relativity is in excellent
Figure 10x GW from Binary Pulsars [view large image]	agreement with observations as shown in Figure 10x. Thus, the observation indirectly confirms the phenomena of gravitational radiation.

	Figure 10y illustrates the merger of two orbiting black holes. Initially, the gravitational signals from such an event would show oscillation with increasing amplitude and decreasing wavelength as the black holes spiral toward each other. A chaotic pattern of gravitational waves may be given off at the moment of merger. Finally, the resultant single black hole is expected to
Figure 10y Black Hole Merger [view large image]	"ring", creating waves with diminished amplitude. This event will emit no x-ray burst, not even a flash of light.

result of the collision between two parallel universes floating in higher dimensional space. Each of these models predicts a specific pattern of gravitational waves emitted from the Big Bang. NASA and ESA plan to launch the Laser Interferometer Space Antenna (LISA) to detect gravitational wave by 2015. It consists of three satellites orbiting the sun (Figure 10za). They will be linked by three laser beams, forming a triangle of light. They are designed to detect a change in their spacing as small as 1/10 the diameter of an atom. With such sensitivity LISA might be able to detect gravitational waves created immediately after the birth of the cosmos. It offers a chance to select between the contesting cosmological models, and also provides an opportunity to test the string theory.

Figure 10za LISA [view large image]

		not yet detected the elusive gravitational wave, but managed to place an upper bound on the SGWB in the frequency band df around the frequency f ~ 100 Hz. The SGWB is defined by the formula: GW = (f / c) (dGW / df)
Figure 10zb LIGO [view large image]	Figure 10zc SGWB [view large image]	where GW is the energy density of gravitational radiation contained in the frequency range df, and c is the critical energy density of the Universe.

Time

For 10 billion years, the universe has been in existence without bothering with the definition of time. It started about 3.5 billion years ago when unicellular organisms took up residence on Earth. They had to adjust their activities according to the daily and yearly cycles. Since then all living beings including human come equipped with biological clocks within to adopt to these rhythms. For thousands of years, protohumans probably had only dim notions of time: past, present, and future. Beginning around 2500 BCE, systemic definitions of time were developed in the form of calendars. The Egyptian pioneers first divided a day into 24 units. Other calendars were linked to religion and the need to predict days of ritual significance, such as the summer solstice. All calendars had to resolve the incommensurate cycles of days, lunations and solar years, usually by intercalating extra days or months at regular intervals. The Julian calendar was established at 46 BCE. The first mean of measuring daily time was probably the Egyptian shadow stick, dating from about 1450 BCE. It was soon followed by the water clock or clepsydra (Figure 11) and the sandglass or hourglass, in which time is measured by the change in level of flowing water or sand. The

Figure 11 Clepsydra [view large image]

earliest mechanical clocks containing movable parts were built about 700 years ago. It had no minute hand.

Instead of trying to define time or provide an answer to the philosophical question of "What is time?", some of the characteristics of time are listed in the followings:


• Thermodynamic arrow of time - The laws of physics do not care about the direction of time, i.e., the mathematical formulations are invariant if the direction of time is reversed. Yet there is a big difference between the forward and

backward directions of real time in ordinary life. We have no difficulty to tell the sequence of events in Figure 12a (from 1 to 9), because it is highly improbable for the crumbling dust to return to the structured building in the reversed direction. It is usually explained by the second law of thermodynamics, which states that in the macroscopic world there is a tendency for a closed system moving toward greater disorder. This is the

Figure 12a Entropy - Thermodynamic Arrow of Time[view large image]

thermodynamic arrow of time in Figure 12b.

Cosmological arrow of time - Since volume expansion tend to increase the entropy of a system, the cosmic expansion seems to be a good indicator for the direction of time (Figure 12b). But what would happen if and when the universe stops expanding and begins to contract? According to Hawking, he used to believe that disorder would decrease when the universe recollapse. Now he realizes that the no boundary condition (see imaginary time) implies that disorder would in fact continue to increase during the contraction. He attributes the association between the thermodynamic and cosmolgical arrows to the no boundary condition instead of volume expansion.

Figure 12b Arrows of Time [view large image]

Asymmetry between past and future - The asymmetry between past and future has two aspects. On the one hand, we know events happened in the past by memory recall, history books, fossil records, and astronomical observations etc., but we know nothing about the future. The direction of time in which we remember the past and not the future is referred to as the psychological arrow of time (Figure 12b). On the other hand, we can only move forward into the future and can never go backward to the past physically. This kind of asymmetry is related to the principle of cause and effect (causality), which is an important concept in physical theories. For example, the notion that events can be ordered into causes and effects is necessary to prevent contradictions such as the grandfather paradox, which asks what happens if a time-traveller kills his own grandfather before he ever meets his grandmother. Within special relativity, causality can be preserved by forbidding information from traveling faster than the speed of light - the worldline is not allowed to loop back to the past. It is strongly suspected that general relativity also preserves causality and forbids agents from changing the past, despite the possibility of developing a closed time-like curve.

Figure 12c Time Machine [view large image]

The idea of using time machine to visit the past is still a hot topic in science fictions. Figure 12c shows a fictitious time machine.

• Rate of change - Time is often equated to the change of a variable dR. Mathematically, it can be expressed as
  dR/dt = constant rate of change, where dt denotes the change in time. One revolution of the Earth around the Sun defines a year. One complete rotation of Earth defines a day. The moving hands of clocks define hour, minute, and second. In these cases, the variable R is the angle of rotation. In short, the units of time are often defined by cyclic motions and their subdivisions. The accuracy in clocks has improved over the last 700 years until today (Figure 13a),

  		an atomic clock known as NIST-7 is accurate to 10-9 second per day. Modern quartz clocks use the piezo-electric properties of the quartz crystal, which vibrates at a specific frequency when placed in an alternating electric current circuit. The induced "crystal current" is amplified and used to operate an LED display or electrically actuated hands (see Figure 13b). Atomic clocks use the frequency of atomic radiation to regulate a quartz crystal clock (see Figure 14a). One second is now defined as the duration of 9,192,631,770 periods associated with the microwave from the hyperfine transition of cesium atoms. All types of design
  Figure 13 a, b Clocks [view large image a, b]	Figure 14a Atomic Clock [view large image] [other image]	depend on the tuning of the microwave cavity to find the frequency, which induces the maximum number of transitions between the hyperfine states.

  This is used as reference to measure error in the output frequency (to the clock). Any deviation of the output frequency (originated within the quartz oscillator) against this standard will be corrected by the Servo Control (see also "other image" under Figure 14a). The process is similar to the adjustment of the hand manually in the conventional wall clock by comparing it to a more accurate time piece.

  In the next generation atomic clock, the microwave cavity is replaced by the frequency comb, which generates a train of million laser pulses each one with a different frequency (in optical range). The frequency comb is used to probe a lattice of atoms cooled to microKelvin temperature. The sweep produced a profile (the bell curve in Figure 14b). Due correction is made according to the difference between the frequencies at the peak (corres-

  Figure 14b Atomic Clock, Next Generation [view large image]

  ponding to the most intense signal induced by one of the comb frequencies) and the actual output to the clock. It is claimed that this new model is ten times more accurate than the conventional atomic clock. One of the

  problems is synchronization, which is more difficult to achieve than the microwave signals from the older model.
  
  
• Imaginary time - The imaginary time T is defined by: t it'. This simple substitution is rather controversial in the community of theoretical physicists. Some argue that it is merely a mathematical trick, or a convenient tool devoid of any physical significance. Others suggest that imaginary time is the true physical quantity. Whatever its merit, imaginary time does provide an alternate computational technique and conceptional viewpoint as shown in the following examples (Note that imaginary and real are just mathematical terminologies in this context. It has nothing to do with the usual connotation of mental perspective.).
  
  Quantization of particles and fields is most elegantly prescribed by the method of path integral. The mathematical formula in term of the real time t for the probability amplitude to go from q(t₁) to q(t₂) is proportional to:
  
  
  
  where the sum is over all paths and L is the Lagrangian (a function of the position and velocity). The oscillating factors in this formula are very difficult to manipulate. However by substituting the real time with the imaginary time, Eq.(22a) is changed into:
  
  
  
  which become the more manageable exponentially decreasing functions. At the end of the computation, the imaginary time can be switched back to the real time.
  
  

  Meanwhile, if the imaginary time is substituted into Eq.(10), it changes into a form familiar to Euclidean geometry: ds2 = dx2 + dy2 + dz2 + c2 dt'2 ---------- (23) While the real time in Eq.(10) restricts the time direction within the light cone, for the imaginary time there is no difference between the time direction and directions in space. Thus according to Hawking, it is possible for such space-time to be finite in extent and have no

  Figure 15a The "No Boundary" Proposal [view large image]

  singularities that formed a boundary or edge. As shown in Figure 15a, space-time would be like the surface of the earth, only with two more dimensions. Such Euclidean sphere has zero points at the North and

  South Poles, but these points would not be any more singular than these Poles on earth. This is the "No Boundary" proposal, which not only avoids the singularity in space but also does away with the initial condition in time. The corresponding universe in real time would have a minimum size, which corresponds to the maximum radius of the history in imaginary time. At later real times, the universe would expand at an increasing inflationary rate to a very large size (see lower right diagram in Figure 15a or Figure 10q). By combining the two examples as illustrated above, a path integral similar to Eq.(22b) can be used to calculate the probability amplitude for the no boundary universes. The sum over

  histories is actually performed backward in time from the current state of the universe such as three-dimensional and flat, then construct the set of all possible histories that would end up like ours (with variables such as inflation, big crunch, ...), and finally sum them up by assigning a weighting factor to each to produce the probability amplitude for the history of a certain universe (see Figure 15b). Although the answer matches observations (i.e., our universe is the most probable)

  Figure 15b Sum over Histories [view large image]

  and incur no singularity, many physicists argue that this is just giving up on the problem of explaining why our universe is the way it is - it is not, they say, science.

  The Schrodinger equation in quantum mechanics has been carried over to cosmology. The only difference being that its solutions, instead of describing the possible values of the position or momentum of a particle, represents the possible geometrical state of the Universe. Quantum effects are thus introduced into cosmology with such interpretation. This is called Wheeler-DeWitt equation: [ d2 / d R2 - U(R) ] = 0 ---------- (24) where R is the scale factor, U is the potential (a function of R, the curvature, the cosmological constant, the densities of matter, and radiation), and is the wave

  Figure 15c Wave Function of the Universe [view large image]

  function of the universe, in which the particle position is replaced by the radius R for a multitude of universes. The result confirms that the expanding isotropic universe similar to our own is the most probable one (see Figure 15c).

  The analogy has been extended further by quantization of the Wheeler-DeWitt wave function leading to the possibility of creating and annihilating universes.
  
  
• The psychological time (subjective time) - The psychological perception of time is affected by such things as medications,

  time of day, level of happiness, external stimuli, and even the temperature. Einstein once remarked, "When you spend two hours with a nice girl, you think it's only a minute. But when you sit on a hot stove for a minute, you think it's two hours." Figure 16 shows Einstein and friend in a 1985 movie called "Insignificance". Thus, the time line experienced by the conscious brain is often quite different from the "objective" time line of events occurring in the real world. Beside the circadian clock, which controls activities in 24-hour cycle, and the millisecond timing, which is involved in fine motor motion; it is found that there is a region of the brain called the striaturm (a cluster of nuclei that includes the putamen and the caudate nucleus), which is used to perceive the passage of time in the seconds-to-hours range. As neurons in the brain regions go about their business, coordinating movement, attention, memory and so on, they produce waves of electrical

  Figure 16 Einstein and Friend [view large image]

  excitation that are detected by the striatum and integrated into an estimate of how much time has passed - producing the subjective time. It is suggested that such subjective time can be manipulated by brain chemistry, in particular the dopamine (one kind of

neurotransmitters that controls arousal levels). It is known that patients with disorders in its secretion, such as Parkinson's disease, Huntington's or schizophrenia, also suffer disturbances in their perception of time. It turns out this is because their neurochemistry - specifically their dopamine level - somehow alters the speed of their subjective internal clock. Schizophrenics have too much dopamine in the brain, their clock is so fast that it feels like the whole world is crazy. Stimulants such as cocaine, caffeine and nicotine make time passing faster, while sedatives such as Valium and cannabis slow it down.


• End of time (Figure 17) - Recently, there is suggestion that time is just an illusion. Time as such does not exist. There is no invisible river of time. But there are things that can be called "instants of time", or "Nows". As we live, we seem to move through a succession of Nows. The idea is that when we think we are seeing actual motion, the brain is interpreting all the simultaneously encoded images and playing them as a movie - frame by frame. Conceptually, it is similar to plot points in the phase space, which is also known as configuration space where the element of time is left out. Figure 18 depicts a very simple configuration space for an universe of three particles A, B, and C. The lengths of each side (of the triangle) form the three grid axes AB, BC, and CA. The seven triangles represent several possible arrangements or NOWs of the model universe. The black diamond indicates the position of each of these triangles or NOWs in the configuration space. In general if there are n particles, the configuration space will be constructed with n grid axes. We have no access to the past and future. The past is only in our memories or some sort of records. They are actually a present phenomena. The future is the not yet realized events equally inacessible. We only experience a continuous sequence of NOWs one after the other. Time was invented for the convenience of human to keep track of

		the changing NOWs. Note that the represent-ation in phase space is background independent. The spatial coordinates x, y, z are absent in the picture, and so is the time. Essentially, this is the foundation of the so-called relational theory in which all that matters is the relationships or links between the events. It plays a crucial role in the formulation of loop quantum gravity, in which space and time are discrete quantities and evolve dynamically like the atoms.
Figure 17 End of Time [view large image]	Figure 18 Configuration Space [view large image]

Thermal Time - Another attempt to do away with time asserts that it is all an illusion. The thermal time hypothesis (Figure 19) suggests that a statistical effect gives rise to the "appearance" of time. Similar to temperature, which is the average of the momentum of each molecule, the same applies to the thermal time but including many more constituents such as space (which is expanding on the average). It predicts that the ratio of the observer's proper time to the thermal time is the surrounding temperature. A toy model has been constructed successfully from the CMBR data to explain the cosmic expansion as described by standard cosmology. It can also reproduce the temperature of a black hole associated with the Hawking radiation. The idea follows the rework of quantum mechanics without time. The evolution in time is replaced by the variation of correlations between things as mentioned above. Instead of "collapsing" the wave function of an electron, both the electron and the measuring device are described by a single wave function, and a single measurement

Figure 19 Thermal Time [view large image]

of the entire set-up causes the collapse. It is anticipated that combining quantum mechanics with general relativity would become less daunting when it is rewritten in time-free form. The loop quantum theory is an example adopting such idea.




Two Times - Instead of playing with "no time", how about constructing theory with 2 dimensions of time? It was found long time ago that such theory leads to negative probability. Worse still, it admits the weird case of time travel backward as shown in Figure 20. Thus, the two-dimensional time gives every appearance of being a non-starter. However, it is shown in a research work in 2007 that the problems disappear if some kind of symmetry is imposed on position and momentum, and the theory involves one more spatial dimension, i.e., our 4-D space-time is generalized to 6-D space-time with 4 spatial dimensions and two time dimensions. It is suggested that the world we see around us is merely a "shadow" of a 6-D world. According to this idea, the standard model is just one shadow (projection) of this 6-D world. There are other shadows that include gravity, finally uniting it with the standard model. Although the effects of an extra time dimension are subtle, it is purported to be very real out there.

Figure 20 Two Dimensional Time [view large image]

For example, the problem associated with the axion (a not yet observed entity) can be resolved by the application of 2D-time physics without such hypothetical particle.