# Special Relativity and Gravity

As Einstein was seeking a relativistic theory of gravity, he thought that special relativity should be upgraded to general relativity thus promoting the Minkowski space to curved pseudo-Riemannian (Lorentzian) one. Does this mean that special relativity as a theory never discussed gravity from any perspective?

• If you are simply asking if special relativity regards gravity? It does not. – Sponge Bob Oct 8 '15 at 16:35
• Yes, then why did be considered a revolutionary theory of its time and was a candidate to replace Newtonian Mechanics before even GR was invented? @SpongeBob – Beyond-formulas Oct 8 '15 at 16:47
• It wasn't. Einstein knew that SR could not model gravity. The importance of SR is that it unified the behaviour of mechanics with the framework of electromagnetism. Previously these were considered incompatible. Also SR accounted for astronomical puzzles like the aberration of light. This is a really big deal. – m4r35n357 Oct 8 '15 at 18:32
• Via the use of Lorentz transform and Minkowski space (which put time and space on an "equal" footing in spacetime) for mechanics. Previously mechanics used the Galilean transform, where time affected space but space did not affect time. – m4r35n357 Oct 8 '15 at 18:51
• The Lorentz transform originated in and describes the world of EM. – m4r35n357 Oct 8 '15 at 19:11

## 4 Answers

Does this mean that special relativity as a theory never discussed gravity from any perspective?

It all hinges on the luminiferous aether which was prevalent in the 19th century theories:

The Michelson Morley experiment was crucial in discovering that there does not exist a luminiferous aether.

The Michelson–Morley experiment was performed over the spring and summer of 1887 by Albert A. Michelson and Edward W. Morley at what is now Case Western Reserve University in Cleveland, Ohio, and published in November of the same year. It compared the speed of light in perpendicular directions, in an attempt to detect the relative motion of matter through the stationary luminiferous aether ("aether wind"). The negative results are generally considered to be the first strong evidence against the then-prevalent aether theory, and initiated a line of research that eventually led to special relativity

To start with the Lorenz transformations were discovered/invented to make consistent Maxwells equations with the existence of a luminiferous ether, i.e. an inertial framework against which everything else would be moving with classical mechanics equations of motion.

Here is the history of Lorenz transformations, the lynch pin of special relativity.

Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous ether hypothesis, were also seeking the transformation under which Maxwell's equations are invariant when transformed from the ether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time"). Henri Poincaré gave a physical interpretation to local time (to first order in v/c) as the consequence of clock synchronization, under the assumption that the speed of light is constant in moving frames. Larmor is credited to have been the first to understand the crucial time dilation property inherent in his equations.

In 1905, Poincaré was the first to recognize that the transformation has the properties of a mathematical group, and named it after Lorentz.

Later in the same year Albert Einstein published what is now called special relativity, by deriving the Lorentz transformation under the assumptions of the principle of relativity and the constancy of the speed of light in any inertial reference frame, and by abandoning the mechanical aether.

The "out of the box" thinking of Einstein comes when he applied the Lorenz transformations to particles, not light. It took some time to confirm it , and the real validation comes from nuclear physics and the huge number of particle physics experiments which can only be interpreted by assuming a four dimensional space time.

As you see from the above precis gravity does not enter into the special relativity validation.,

General relativity was not the first gravitational theory proposed for special relativity. The first steps were just to treat it like any field theory (like it is in classical physics).

The most basic is the scalar field theory, which has been proposed in some variations by Einstein, Abraham and Nordström :

$$\square \Phi = 4 \pi G \rho$$

Which is just the Newtonian Poisson equation with $\Delta \rightarrow \square$. To remain Lorentz invariant, $\rho$ is not the energy density but the rest mass density.

Nordström also made later on the following theory :

$$\Phi \square \Phi = -4 \pi G T$$

With $T$ the trace of the stress energy tensor.

While both agree with the Newton equation in the classical limit, they has a few problems. Light is not deflected by gravity (since it has no rest mass or a trace of the stress energy tensor), and the precession for Mercury is of the wrong sign and magnitude. It is also a bit odd to put into Lagrangian form, since the stress energy tensor is directly in it.

Another proposed theory was the tensor theory, which was

$$\frac 1 2 \square h_{\mu\nu} + \partial_\mu \partial_\nu h - \partial_{\{\nu}\partial^\sigma h_{\mu\}\sigma} + \eta_{\mu\nu} (\partial^\alpha\partial^\beta h_{\alpha\beta} - \square h) = \frac \kappa 2 T_{\mu\nu}$$

Which is basically the same as linearized GR. Due to this, it has a pretty good agreement for experimental values, but it has theoretical problems that if the matter fields are on shell, energy is not conserved any more. To fix this, the procedure is to add more and more counter terms, which will leave you with general relativity in the end.

• what do you mean by matter fields are "on shell"? – Beyond-formulas Oct 10 '15 at 13:01
• On shell meaning obeying the equation of movement of the matter field. – Slereah Oct 10 '15 at 13:15
• I am also interested in why gravity does not fit into Special Relativity. Is it because it makes the result uglier? Or it simply would introduce a contradiction (to some held beliefs)? Unfortunately, there have been too many rumors about SR/GR since they were made popular, making it so hard to get the real information. Would you mind pointing out some reference for your answer? Or, even better, do you know some discussions about this that are worth reading? Thank you very much. – Student Mar 2 '19 at 3:13

Einstein considered a theory of gravity involving a scalar field in a flat spacetime. Einstein discarded this theory because he thought it broke conservation of energy. Domenico Giulini has claimed that Einstein's reason for discarding the theory was wrong but the theory conflicts with experiment anyway:

https://arxiv.org/abs/gr-qc/0611100

see H.Poincaré 1905 Sur la dynamique de l’électron

his § 9. — Hypotheses on gravitation is very, very, on top of what you are asking.

IMO, his document is much more pertinent to Relativity than the Einstein's one and he deals with gravity under SR principles. He could not make it complete because he did not know the speed of the solar system thru the background, as he confessed. (He needed one more equation to close the Mercury' issue).