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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 for electromagnetic wave 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: x ^{2} + y^{2} + z^{2} = c^{2} t^{2} (for an observer in the S frame) ---------- (6)x' ^{2} + y'^{2} + z'^{2} = c^{2} t'^{2} (for an observer in the S' frame) ---------- (7)
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## Figure 02c Constant Speed of Light |

x' = (x - Vt) / (1 - V

The inverse transformation is:

x = (x' + Vt') / (1 - V

Eq.(8a) reduces to the Galilean transformation in Eq.(4) when V/c << 1. Figure 02c shows pictorially the wave fronts in the S and S' frames from the perspective of an observer in the rest frame. In this way, x, y, z, and ct form a four-dimensional space known as the Minkowski space-time. It revolutionizes high-energy physics when velocity of the particles is close to the velocity of light (relative to the observer's inertial frame). All the basic formulae such as the field equations and the Lagrangians have to be invariant under the Lorentz transformation, and the definition for many physical entities changes form at high speed as listed in the followings:

- 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 = t_{o}. According to the first formula in Eq.(8a), the relation is given by: L = x_{2}- x_{1}= L' (1 - V^{2}/c^{2})^{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). - 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 = t_{2}- t_{1}=

T' / (1 - V^{2}/c^{2})^{1/2}or T' = T (1 - V^{2}/c^{2})^{1/2}. The effect is known as time dilation and T' is referred to as the proper time - the time read by a clock moving together with the 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 - V ^{2}/c^{2})^{-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-lines are different, and not interchangeable, as there is no inertial frame in which the traveling twin is always at rest.#### Figure 04a Dilation [view large image]

#### Figure 04b Twin Paradox

_{}

ds- 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 = t
_{o}at x = x_{1}and x = x_{2}, are separated by a time interval t'_{2}- t'_{1}= (V(x_{1}- x_{2})/c^{2}) / (1 - V^{2}/c^{2})^{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:

c^{2}dt^{2}- dx^{2}- dy^{2}-dz^{2}= 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 d *l*in Eq.(5) under the Galilean Transformation, it can be shown that for object moving below the speed of light, the space-time interval:^{2}= c^{2}dt^{2}- dx^{2}- dy^{2}- dz^{2}---------- (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 v^{2}= [(dx/dt)^{2}+ (dy/dt)^{2}+ (dz/dt)^{2}], Eq.(10) can be re-written as: ds^{2}= (1-v^{2}/c^{2})c^{2}dt^{2}.The interval is called time-like if ds ^{2}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 ds^{2}< 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 if event 1 moves up to event 1a. The interval ds plays the role of the time parameter in Newtonian mechanics to keep track of the development of events such#### Figure 05 Minkowski Spacetime Transformation [large image]

as in the generalized equation of motion in Eq.(12b). Note that the velocity of light c is constant in all Lorentz frame of references in Figure 05 as originally envisioned.

- 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 = t
- 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 (don't ever try to visualize the geometric configurations in such pseudo-Euclidean space - it will drive you crazy). 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 - V^{2}/ c^{2})^{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/c^{2}) 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

u_{4}= icdt/ds = i.

By definition the square of the 4-velocity u^{2}= -1 is invariant under the Lorentz transformation. - Similarly the 4-momentum is defined as:

p_{}= m_{o}v_{},

and p_{4}= im_{o}c.

The square of the 4-momentum p^{2}= -m_{o}^{2}c^{2}is another 4-scalar quantity invariant under the Lorentz transformation.

The 4th component is linked to the total energy E by the formula:

E = -ip_{4}c = m_{o}c^{2}= mc^{2}

where m = m_{o}. The relationship is justified by the correct limit at low speed when E m_{o}c^{2}+ m_{o}v^{2}/2. This is the origin of the most celebrated formula_{}derived by Einstein. - Thus, the mass for a particle moving at velocity v is given by m = m
_{o}/ (1 - v^{2}/c^{2})^{1/2}, where m_{o}is the rest mass (relative to the observer's rest frame). This formula shows that m as v c. It means that 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 c
^{2}or E = m_{o}c^{2}+ 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 - v^{2}/c^{2})^{1/2}-1] m_{o}c^{2}. - In term of the 3-momentum
**p**, E = (m_{o}^{2}c^{4}+**p**^{2}c^{2})^{1/2}from which the relationship between v and**p**can be derived in the form :

(v/c)^{2}= 1/[1 + (m_{o}c/**p**)^{2}] or**p**^{2}= m_{o}^{2}c^{2}{(v/c)^{2}/[1 - (v/c)^{2}]}

For m_{o}^{2}> 0, the particle is always moving slower than the velocity of light. For

m_{o}^{2}= 0, v is always equal to c. If m_{o}^{2}< 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 |m_{o}|c (see Figure 06). - 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 = |d**x**/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_{o}^{2}c^{2}= E^{2}/c^{2}-**k**^{2}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 ds ^{2}= c^{2}dt^{2}- d**x**^{2}0ds ^{2}= c^{2}dt^{2}- d**x**^{2}< 0v > c - tachyon 4-Momentum k = ( **k**, iE/c)k = ( **k**, iE/c)Outside light cone 4-Momentum Squared m _{o}^{2}c^{2}= k^{2}= E^{2}/c^{2}-**k**^{2}0m _{o}^{2}c^{2}= k^{2}= E^{2}/c^{2}-**k**^{2}< 0Off mass-shell - tachyon 4-Momentum Transfer k' - k = [( **k**' -**k**),*i*(E' - E)/c] = qk' - k = [( **k**' -**k**),*i*(E' - E)/c] = qOutside light cone 4-Momentum Transfer Squared (k' - k) ^{2}= (E' - E)^{2}/c^{2}- (**k**' -**k**)^{2}= q^{2}0(k' - k) ^{2}= (E' - E)^{2}/c^{2}- (**k**' -**k**)^{2}= q^{2}< 0Off 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 a complex number. - In special relativity the equation of motion for a particle is:

f_{i}= m_{o}c^{2}(du_{i}/ds)

where the space component of the velocity is**u**= (**v**/c)/(1-v^{2}/c^{2})^{1/2}, the time component is u_{4}= i /(1-v^{2}/c^{2})^{1/2}, while the space components of the four vector f_{i}form a vector**f**/ (1-v^{2}/c^{2})^{1/2}, the time component is f_{4}= i**f**(**v**/c) / (1-v^{2}/c^{2})^{1/2}. These equations are 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 (as these frames are only artificial objects). 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.

## Figure 03 Lorentz Contraction |

## Figure 06 Mass and 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 |

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