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ds

This is known as the Schwarzschild solution. The integration constant 2GM/c

- As r , the space-time metric reduces to the expression for flat space-time:

ds^{2}= c^{2}dt^{2}- dr^{2}- r^{2}(sin^{2}d^{2}+ d^{2}) ---------- (14)

- At the non-relativistic limit where v
^{2}/c^{2}<< 1, an expression for the space-time metric can be derived from classical mechanics:

ds^{2}= (1 + 2*V*/c^{2}) c^{2}dt^{2}- dr^{2}- r^{2}(sin^{2}d^{2}+ d^{2})

where*V*is the gravitational potential.

Comparing with the space-time metric g_{44}in Eq.(13) in the region where r > 2GM / c^{2}, the Newton's law of gravitation:*V*= - GM / r is recovered from general relativity. Alternately the force of gravity F can be expressed in term of the metric tensor as: F = -GMm/r^{2}= -(mc^{2}/2)(dg_{44}/dr), which provides the link between physics and geometry explicitly. This relationship also shows that at the non-relativistic limit only the time dilation effect (relating to g_{44}) is important. The effect of spatial curvature (relating to g_{11}) is negligible at this limit. - At the distance where r = r
_{s}= 2GM/c^{2}, 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 r
_{s}is outside the central object. For the Earth, and the Sun, r_{s}is well inside the physical boundary (r_{s}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 2-D circles are the projection of 3-D spheres - the hyperspace. The vertical linrd denote the "stretch" of space as a function of the radius r. The slope of the curve can be considered as 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 - r_{s}/r)^{1/2}

where r is the distance of the light source with respect to r_{s}, and the observer is assumed to be at infinity. The redshift becomes infinity when r = r_{s}. This is the effect that makes light invisible with the source at r_{s}. - 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 is the time interval recorded by an observer plunging toward the black hole and T _{o}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 r_{s}. 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 r_{s}is equal to 2GM / r_{s}^{3}= c^{6}/(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 r_{s}from the center of a black hole. The mass of the Earth M 6x10^{27}gm giving r_{s}1 cm. With the Earth's surface at about 0.63x10^{9}cm, and the GPS satellite about 2.65x10^{9}cm from the center, the % difference of clock rate between the two locations is about 50x10^{-9}(with respect to T_{0}), which is one of the corrections that has to be made in order to produce an accurate map reading.#### Figure 09d Tidal Force

_{}#### Figure 09e Time Dilation

_{}

- Inside r
_{s}, 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 g_{44}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; and it might provide a gateway leading to another universe. Figure 09f shows a collapsing star, and the behavior of the light cone 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 subtends an angle of 45^{o}as in flat space. Since the effect of gravity is to retard the outward motion and to advance the inward motion, the angle shrinks and tilts toward the black hole for light cones closer to the center. The retardation of the out going speed of light is given by the formula: v = c(1 - r_{s}/r). At the event horizon v = 0, not even light can escape to outside. The directions of the cone only point to the center once inside the black hole, the movement of all objects approach the speed of light progressively near the singularity, there is no turning back.#### Figure 09f Black Hole, Inside [view large image]

d = ± (g- 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]c ^{2}d^{2}= ds^{2}= g_{44}c^{2}dt^{2}

where g_{44}is positive according to the convention. Thus,_{44})^{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. - Inside r
- 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 / c ^{2}= r_{s}. The throat of the wormhole is located at r = b(r) = r_{s}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: d *l*^{2}= dW^{2}+ dr^{2}+ r^{2}(sin^{2}d^{2}) = dr^{2}/ (1 - r_{s}/r) + r^{2}(sin^{2}d^{2}) ---------- (15b)

Eq.(15b) can be used to equate dW = (r/r_{s}- 1)^{-1/2}dr. Integration of the equation gives W^{2}= 4r_{s}(r - r_{s}), which is a parabola function with vertex at W = 0, r = r_{s}. 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, g_{11}and g_{44}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 becomefeasible. 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/c ^{2}), 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]

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## Figure 09a Black Hole, 3-D Spatial Curvature [view large image] |
## Figure 09b |

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## Figure 09c Redshift |
On the other hand if the positions of the source and observer are switched, the wavelength will be blue shifted according to the formula: = (1 - r_{s}/r_{ob})^{1/2} _{o}. There would be no change of wavelength if both observer and source are falling together into the black hole. |

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