General Relativity Appendix

The Principle of Equivalence:

The local laws of physics are equivalent for all free-floating observers.

With this extension of the equivalence principle postulated in special relativity, Einstein introduced perfect symmetry into the physical universe.

The Bending of Light

Ingredients:

One Eqivalence Principle,
one physicist isolated in intergalactic space,
and one stationary physicist in gravitational field.

1) Place each physicist in an indentical rocket(i.e. metal box)
2) Bring the physicist in space to a constant acceleration exactly equivalent in magnitude to the gravitational acceleration experienced by the stationary physicist.
3) Coerce each physicist to shoot photons horizontally (normal to the direction of acceleration) from one wall of their box to the other wall.
4) Notice that the accelerating physicist will always observe the photons to hit the opposite wall in a lower height than the one emitted at.
5) To keep things from coming out half-baked, carefully apply the Equivalence Principle to conclude that the physicist in the gravity field must also see his photons hit the wall lower than the height they started.

A approximation of the curvature near Earth can be done using elementary trigonometry that results in an angular deflection of:

where g is the local magnitude of acceleration/gravity, d is the horizontal distance traveled by the light, and c is the speed of light. This effect usually amounts to deflection of less than 1 billionth of an arcsecond. Near black holes, however, the weak-field approximations used to derive the above equation cannot be used and the deflections correctly calculated are by no means negligible.

Gravitational Lensing

Since light must follow curved paths near massive bodies, the apparent origin of light rays can be greatly affected. In the exaggerated figure to the right, light from the blue object passes near a massive object, such as a black hole, and is deflected towards the eye so that it appears to originate from the 'virtual' lavender object. If there is a smaller luminous object, such as a quasar, directly behind a massive object, perhaps black hole but most likely something like a galaxy, the light from the smaller object that might've passed out of our field of view, can pulled and focused into apparent images of the original object. The most famous of these gravitational lenses is the Einstein Cross. (The four outer blobs on the cross are the virtual images of one quasar behind a central galaxy.

Gravitational Redshift

Application of the Equivalence Principle as above in "The Bending of Light" to the scenario of photons directed radially along the classical field results in another prediction, that of gravitational frequency shift. This effect is mostly thought of in terms of redshift, since light leaving the areas of high curvature is of most interest and is redshifted. The equation which exactly describes frequency shift is below. If if photon is emitted at a distance r0 from a body of mass M, then the photon's frequency, at a far distance where the curvature is negligible, is shifted from its emitted frequency by the factor under the radical. Near Earth, as above, the effects of space-time on frequency are hardly observable(the frequency shift is on the same order as the angular deflection). Redshift near massive bodies is a different story. For all bodies, a photon emitted from the near Schwarzchild radius has its frequency reduced to almost zero. From the exact radius, the frequency observed is zero exactly; no light comes from Schwarzchild radius. The reason redshift is not observed in common experience is because of the small size of the Earth's Schwarzchild radius with respect to the surface of the Earth.