How (or why) equivalence principle led to Einstein field equations?

Question

If equivalence principle was origin of general relativity what was the process that this principle led Einstein to developed his theory of general relativity?

Answer 1

In 1921 Einstein gave a series of lectures in Princeton, that you can read today under the title "The Meaning of Relativity". It is an early and very special description of General Relativity, where he emphasizes much the concepts and reasoning that lead him to the theory.

Nobody is able to know what was really inside Einstein's mind, but in that lectures he acts as if he had invented General Relativity through sort of a heuristic process based on the Equivalence Principle. If he did that in these lectures for others to better understand him, or if he really used the Equivalence Principle as the origin of General Relativity, it is hard to tell, but I would bet it is the second case. Other ideas like Mach's Principle (which is another question that, incidentally, remains still unsolved) are present in that lectures as well.

It is a very nice book that shows a little of how Einstein thinking worked (at least it gives that illusion). However it is not a popular description (anyway it is easier to read than the 1916 paper). I will try to offer a quick summary here of how that lectures lead from the Equivalence Principle to General Relativity, although this attempt is nearly a blasphemy and you should try having a look at the Einstein text (there is nothing better). Note that I will quote some paragraphs of the book. I can do that because the copyright of that lectures has expired, they are offered for free for instance in the site of the Gutenberg Project. I will also insert short explanations of basic things that you probably know, for other readers.

So, according to that book, these are more or less the steps connecting the Equivalence Principle and the Field Equations:

1: Invariance of the infinitesimal interval, extended to non-inertial observers

Special Relativity had dealt already well with phenomena in the absence of gravity. In that theory, the distance measured between two infinitely close events in spacetime, is the same for two observers $O$ and $O'$ who move with constant velocity with respect to each other:

$$cdt^2-dx^2-dy^2-dz^2 =cdt'^2-dx'^2-dy'^2-dz'^2 $$ You may think of an "event" as a spark: something instantaneous happening at a point in space and time.

The quantity which is directly measurable by our unit measuring rods and clocks, $${dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2} $$ is therefore a uniquely determinate invariant for two neighbouring events

The expression below can be written in a shorter form, as in equation (55) of the book:

$$ds^2=g_{\mu \nu}dx^{\mu \nu}$$

And thus, $ds^2=ds'^2$, both observers measure the same interval

What relates $(t,x,y,z)$ and $(t',x',y,',z')$ is a Lorentz Transformation. By means of a Lorentz Transformation, you may calculate for instance the differences in the length of a rod that two differently moving (inertial) observers will measure and therefore the different $dx^{\mu}$. But the set of coefficients $g_{\mu\nu}$ is the same for both observers in Special Relativity, simply a set of four numbers $+1,-1,-1$ and $-1$

2: The new role and tensor character of the $g_{\mu \nu}$ coefficients

The problem was that the Lorentz transformation only works for non-accelerating observers. But the Equivalence Principle stated that an accelerated observer should be by all means equivalent to a non-accelerated observer embedded a gravitational field, and then the brilliant insight of Einstein was to "mend" Special Relativity NOT by trying to re-invent the Lorentz Transformation, but rather by introducing the new information of the gravitational field/acceleration into the $g_{\mu\nu}$ quantities, that had been until the moment merely a set of passive constant numbers independent of the observer.

But, why in the $g_{\mu\nu}$? Well, it was known that relative velocity alters the measures of length between observers. That is equivalent to changing the distance between the axis ticks of one of the observers. If there was a way of doing that kind of shortening, in a more sophisticated way to account for accelerations, it was by altering the coefficients $g_{\mu \nu}$ that multiply the $dx^{\mu}$ in an observer-dependent way, so that the final expression of $ds^2$ remained independent of the observer. This may seem easy to understand now, but in order to have that brilliant idea first, you have to be... well, Einstein!

Therefore, he started searching some way of doing this alteration of the $g_{\mu \nu}$ in order to include accelerations, and so he started looking for its mathematical properties. And the starting point was to demand that the Equivalence Principle holds also in the new situation:

It follows from the invariance $ds^{2}$ for an arbitrary choice of the $dx_{\nu}$, in connexion with the condition of symmetry consistent with Eq (55), that the $g_{\mu\nu}$ are components of a symmetrical co-variant tensor (Fundamental Tensor).

3: Geodesic Postulate connecting the $g_{\mu \nu}$ and gravity

Since $ds^2$ accounts for a short displacement in spacetime, the next step was seeing what happened when putting a short displacement after the other walking always forward, without allowing any deviation of the straight line until building a finite "straight" trajectory. This is important, because it tells you how is the space(time) in which you are moving. For instance, if you start walking in a straight line towards the North Pole, no matter how straight you have walked, when you reach the North Pole you realize that, in 3D space, your trajectory has made a giant arc, and so you know that you have been walking on the curved surface of the Earth.

That kind of "straight" trajectory it is called a geodetic line and mathematically it was already known for "ordinary" geometry of curved surfaces:

A line may be constructed in such a way that its successive elements arise from each other by parallel displacements. This is the natural generalization of the straight line of the Euclidean geometry. For such a line, we have $$\delta \left(\frac{dx_{\mu}}{ds}\right) = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}$$ The left-hand side is to be replaced by $\dfrac{d^{2} x_{\mu}}{ds^{2}}$

We get the same line if we find the line which gives a stationary value to the integral $$\int ds\quad\text{or}\quad \int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}} $$ between two points (geodetic line).

Now you see that, if you were able to know the $g_{\mu \nu}$ an observer "feels" at each point of spacetime, you could derive the shape of the geodetic line between two arbitrary events. Sort of conversely (but not completely in the general case), if some physical principle could tell us the shape of the geodetic line, we would have the connection between $g_{\mu \nu}$ and gravity for each observer. The physical principle needed was another brilliant insight, the Geodesic Postulate:

A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this is a real straight line. The natural, that is, the simplest, generalization of the straight line which is plausible in the system of concepts of Riemann's general theory of invariants is that of the straightest, or geodetic, line. We shall accordingly have to assume, in the sense of the principle of equivalence, that the motion of a material particle, under the action only of inertia and gravitation, is described by the equation,

$$ \frac{d^{2} x_{\mu}}{ds^{2}} + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0$$

In fact, this equation reduces to that of a straight line if all the components, $\Gamma_{\alpha\beta}^{\mu}$, of the gravitational field vanish.

You see that at this high of the book he already calls "gravitational field" to that coefficients (Christoffel Symbols) that account for the way of "shrinking" the observers axis.

To sum up until now:

• In the absence of gravity, a free particle moves in a straight line according to Galilean inertia.
• That free particle is seen moving in a straight line by all inertial observers (restricted Equivalence Principle version of Special Relativity)
• This is extended to spacetime with gravity or accelerated observers, by postulating that free particles are "seen" in spacetime along geodetic lines, by all observers, inertial or not.

4: Newton's $F=\dot p$ from the geodesic equation

Immediately after stating the Geodetic Postulate, he demands that the equations of motion must reduce to that of Newton in the absence of gravity, and so he already finds Newtonian equations of motion:

How are these equations connected with Newton's equations of motion? According to the special theory of relativity, the $g_{\mu\nu}$ as well as the $g^{\mu\nu}$, have the values, with respect to an inertial system (with real time co-ordinate and suitable choice of the sign of $ds^{2}$), $$ \left. \begin{array}{*{4}{>{\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & +1 \end{array} \right\}$$

The equations of motion then become $$ \frac{d^{2} x_{\mu}}{ds^{2}} = 0. $$ We shall call this the ``first approximation'' to the $g_{\mu\nu}$-field.

The "second approximation" consists of adding a small perturbation to that $g_{\mu\nu}$ field that leads him to the Newtonian Second Law $F=\dot p$, which allows to identify that small perturbation he has added with the classical gravitational potential (enclosed in the force, equal to the gradient of the potential). He already has a theory that resembles Newtonian gravity, but only if the gravitational potential (or the acceleration involved) is small.

5: Heuristic search from the general form of the equations of gravity, from resemblances to Poisson Equation and the need for local conservation of Energy-momentum for all observers

So the final part of the way is extending this to a general case. He wants his equations to be similar to that of Newtonian gravity when written à la Poisson, where you can see a position dependent density field in one side, as a source of the "sensible" effects (the gravitational potential) that lie at the other side of the equal sign. He tries that because he already has some object to put in the place of the mass, an object he takes from Special Relativity that describes mass and energy, the Energy-Momentum tensor:

We must next attempt to find the laws of the gravitational field. For this purpose, Poisson's equation, $$ \Delta\phi = 4\pi K\rho $$ of the Newtonian theory must serve as a model. This equation has its foundation in the idea that the gravitational field arises from the density $\rho$ of ponderable matter. It must also be so in the general theory of relativity. But our investigations of the special theory of relativity have shown that in place of the scalar density of matter we have the tensor of energy per unit volume. In the latter is included not only the tensor of the energy of ponderable matter, but also that of the electromagnetic energy.

And now it comes the final part: something analogous to the left-hand side of Poisson equation. If on the right it must be the Energy-Momentum tensor, then the left-hand side must be a tensor too. He struggled for a long time to find it (the so-called Einstein tensor) and even he published a restricted version first, in 1915, that quickly turned out not to be valid in all situations.

To maintain the analogy with Poisson equation, the Einstein tensor had to have certain properties:

If there is an analogue of Poisson's equation in the general theory of relativity, then this equation must be a tensor equation for the tensor $g_{\mu\nu}$ of the gravitational potential; the energy tensor of matter must appear on the right-hand side of this equation. On the left-hand side of the equation there must be a differential tensor in the $g_{\mu\nu}$. We have to find this differential tensor. It is completely determined by the following three conditions:

1.It may contain no differential coefficients of the $g_{\mu\nu}$ higher than the second.

2.It must be linear and homogeneous in these second differential coefficients.

3.Its divergence must vanish identically.

The first two of these conditions are naturally taken from Poisson's equation.

The third condition comes from something analogous to the classical conservation of energy and momentum, which translates into the statement that the divergence of the Energy-Momentum tensor must vanish, again, for all observers:

According to our previous results, the principles of momentum and energy are expressed by the statement that the divergence of this tensor vanishes. In the general theory of relativity, we shall have to assume as valid the corresponding general co-variant equation.

(Bear in mind however, that this is only a local conservation, but this is another question).

There are other forms the Einstein tensor could have in order to satisfy these conditions, and it is interesting to note that, in choosing the final form, it is said that he had sort of an aesthetic sense of simplicity. Well, here is the final step:

Since it may be proved mathematically that all such differential tensors can be formed algebraically (i.e. without differentiation) from Riemann's tensor, our tensor must be of the form $$ R_{\mu\nu} + a R g_{\mu\nu} $$ in which $R_{\mu\nu}$ and $R$ are defined by Eq (...) Further, it may be proved that the third condition requires $a$ to have the value $-\frac{1}{2}$. For the law of the gravitational field we therefore get the equation $$ R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu} $$

The constant $\kappa$ is found to be proportional to the Newtonian $G$ a couple of pages later, by identification with Newtonian gravity in the limit of small energies.

Answer 2

The equivalence principle isn't the origin of general relativity in the sense that you start with the equivalence principle and several pages of maths later end up with general relativity. It is more of a guiding principle. If you accept that the equivalence principle is true, it places restrictions on the sort of theory that could describe gravity.

No-one but Einstein can be sure of exactly how he arrived at GR. From reading various histories of the time it seems to me that once Einstein had come up with the equivalence principle he started looking around for theories that embodied it. There had already been suggestions that gravity could be a geometrical property, but until Einstein and Grossmann hit upon the idea of using Riemannian geometry no-one had made the idea work. Einstein must have realised very quickly that a geometrical theory offered a natural way of incorporating the equivalence principle.

The Development of gravitation theory section in the Wikipedia article on the equivalence principle has more on this. If you're interested in more detail on the history of relativity I recommend Abraham Pais' book Subtle is the Lord.