- #1
Imager
Gold Member
- 112
- 61
How do you create a link to a specific post in PF? I want to link to Matterwave's post 25. The best I could come up with is this to the page...
https://www.physicsforums.com/threa...-energy-stress-curves-spacetime.956580/page-2
https://www.physicsforums.com/threa...-energy-stress-curves-spacetime.956580/page-2
Matterwave said:If you are looking for a nice derivation of General Relativity from first principles like you can derive special relativity from 1. The principle of relativity and 2. the constancy of the speed of light, you won't find one nearly that clean.
In GR though, you do have motivating principles that "hint at" the EFE's but none through which you can "derive" the EFEs. The principle which most motivates the EFEs is Einstein's principle of equivalence.
Your statement of "equivalence of inertial mass and gravitational mass" is the weak principle of equivalence (it's the one mostly experimentally tested for). The Einstein principle of equivalence goes beyond that statement to say essentially that there is no physical way to determine the difference between a locally free-falling reference frame and a reference frame which is truly free in the vacuum of space away from any gravitational fields (other than the gravity of the laboratory itself - i.e. not accounting for self-gravitating effects). In other words, if I stuck you in a small room that you couldn't see or hear or otherwise send signals out of, you couldn't, even in principle, tell whether you were free-falling toward's Earth at 9.8m/s^2 or whether you were floating out in the middle of space (at least, not until you crashed onto the Earth's surface - when your motion is no longer inertial).
The Einstein principle of equivalence says that not only do all bodies fall at the same rate regardless of their mass, but all physics behaves the same way in these differing reference frames. For example, light, even though it's massless, must still be affected by gravity. If, for example, massless particles are not affected by gravity, then one could differentiate the two differing situations by shining a laser (or a grid of lasers) and seeing if that laser (or a grid of lasers) curved away from you. This equivalence principle therefore suggests that it is in theory possible to describe gravity not as a force that acts on "gravitational mass", but as an inherent curvature in space-time. All material bodies are affected by gravity - not just those that have some "gravitational charge". This is very distinct from the case of e.g. the Electromagnetic force where neutrally charged bodies (that also have no magnetic dipole moment) are not affected by electromagnetism.
Einstein's EFE's which relate (the trace-reverse) Ricci curvature to stress energy directly are a beautiful manifestation of the Einstein equivalence principle, but it is not the unique possible theory that incorporates this principle. One can construct other theories which respect the Einstein principle of equivalence (e.g. Brans Dicke theory), but perhaps Einstein's theory is the most "simple" and elegant of them (this last statement is opinion based).
As a technical aside, there is also a strong principle of equivalence which is even more restrictive than Einstein's (in technical terms, it specifies that the space-time metric is the only field which can affect the space-time curvature). Apparently Einstein's General Relativity is thought to be the only theory which satisfies the strong principle of equivalence - but I am not certain of there being any proof of the fact.
... Carry on.