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Usually when we solve field equations, we start with a stress energy tensor and then solve for the Einstein tensor and then eventually the metric. What if we specify a desired geometry first? That is, write down a metric and then solve for the resulting stress energy tensor?

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    $\begingroup$ Usually when we solve the field equations, we start with a stress energy tensor and then solve for the einstein tensor and then eventually the metric. Not really. You can't start by writing down a stress-energy tensor, because to express it in components, you would already need to have some coordinate charts that are adapted to your geometry -- which you don't know yet. What might be more accurate to say is that often we start with an equation of state. $\endgroup$ – Ben Crowell May 30 at 12:52
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    $\begingroup$ If Einstein doesn't deserve to have his name capitalized, who does? $\endgroup$ – David Richerby May 30 at 14:54
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    $\begingroup$ Ben Crowell makes a good point. I'm starting to notice that this misconception comes up here almost once a month. Solving GR with matter involves more than solving EFE from a stress tensor: the full coupled gravity-matter equations, including dynamical equations for the matter fields, must be solved. I think people are confused because often only vacuum solutions and solutions assuming simple forms of the metric (i.e. perfect fluid) are treated in introductory material. $\endgroup$ – Joe Schindler May 30 at 19:41
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    $\begingroup$ @DavidRicherby - Having ones name formally lowercased is the ultimate form of praise in physics. To this end, the einstein is a mole of photons. $\endgroup$ – David Hammen May 30 at 20:17
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    $\begingroup$ @DavidHammen As joule be aware, ampere-fectly aware of that, but this isn't watt's going on, here. ohm my god, this hertz. I'd better stop farad day or two. $\endgroup$ – David Richerby May 30 at 20:26
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You can certainly do this, and indeed it is regularly done. For example Alcubierre designed his FTL drive by starting with the metric he wanted and calculating the required stress-energy tensor. It is a straightforward calculation - it is somewhat tedious to do by hand but Mathematica would do the calculation in a few seconds.

The problem is that the resulting stress-energy tensor will almost always contain contributions from exotic matter, as indeed the Alcubierre stress-energy tensor does, and that means it won't be physically meaningful. The chances of solving the Einstein equation by guessing geometries and ending up with a physically meaningful stress-energy tensor are vanishingly small.

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In most situations the distinction “matter first, geometry second” or “geometry first, matter second” is not that clear cut. Often assumptions are made that constrain both the geometry and stress energy tensor.

Take for example the Schwarzschild metric. We derive it by writing down the most general metric compatible with restrictions imposed by physical description: “isolated static body with spherical symmetry”: $$ ds^2=-A(r)dt^2 + B(r)dr^2 + r^2(d\theta + \sin^2 \theta d\phi^2). $$ Only then we substitute this metric into vacuum Einstein equations (with zero stress energy tensor) and obtain a couple of ordinary differential equations for functions $A(r)$ and $B(r)$. So, we solve equations for a given matter content, but these equations are in a simple form because we specified large parts of geometry first.

Another class of examples are what could be called “science fiction geometries”: time machines, warp drives, traversable wormholes that challenge our intuition on what is allowed in the universe. Such “solutions” are often start from geometry written down with a desired properties but the Einstein field equations are still considered in order to constrain what form of “exotic matter” is needed to obtain such geometries. Parameters of the geometry are often varied in order to minimize the “unnaturalness” of the resulting stress energy tensor. A few examples:

  • Alcubierre warp drive and its variations allows faster than light travel with the help of negative mass.

  • Traversable wormholes would allow travel (or communication) between distant regions of the Universe (or between different universes). See this paper for an example of obtaining conditions of stress-energy for such a spacetime.

Yet another group of examples which have priority of the geometry over matter comes from astrophysics: observations often give us information about spacetime which could then be used to deduce the matter content. That is essentially how $\Lambda$CDM model appears, the matter content, most notably the dark energy is deduced from spacetime structure.

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You can certainly go from the metric to the energy-momentum tensor, but then you’re not “solving” anything. There are no differential equations to be solved if you’re doing that. It’s just a straightforward, although often tedious, computation (of the Einstein curvature tensor, which is proportional to the energy-momentum tensor) that requires nothing more than differentiation and algebra.

It’s not a particularly useful thing to do. Trying lots of metrics and seeing what density and flow of energy and momentum they correspond to doesn’t really give you insight. It’s generally the energy-momentum tensor that is simple, and the metric that’s complicated, so you need to start with the former and solve for the latter.

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This is sometimes jokingly called Synge's method. Here's an excerpt from Ingemar Bengtsson's A Second Relativity Course describing it (see Chapter 5):

We would now like to see a solution describing a physical system that approaches (in some sense) the Schwarzschild solution as it evolves. This can be obtained by means of a method invented by the Irish relativist Synge. Synge’s method is as follows. To solve $$ G_{ab} = 8 \pi T_{ab}, $$ rewrite as $$ T_{ab} = \frac{1}{8 \pi} G_{ab}, $$ choose any metric tensor $g_{ab}$, compute its Einstein tensor $G_{ab}$, and read off the stress-energy tensor $T_{ab}$ from Eq. (5.2). The result is a solution of Eq. (5.1). To avoid any misunderstanding, Synge meant this as a joke (and he did not predict dark matter). A stress-energy tensor computed in this way is not likely to obey any of the positivity conditions that are necessary for it to qualify as physical.

Very occasionally the method works though.

(Bengtsson then proceeds to describe the Vaidya solutions, which are found by basically writing down a metric that looks vaguely like a time-dependent Schwarzschild solution and then interpreting it.)

It's possible that Synge describes his "method" in his 1960 book—the textbook I'm drawing from cites it in the passage above—but I don't have a copy handy.

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  • $\begingroup$ This method could be fun. I wonder what the following metric could give, for the energy-momentum tensor:$$ds^2 = dt^2 - \frac{a^2}{(1 + k(t) \, r^2/4)^2}(dx^2 + dy^2 + dz^2),$$ which is a modification of the Robertson-Walker metric (cosmology), in isotropic form. The scale factor $a$ could be a constant, and the curvature parameter $k$ is now a function of time. $\endgroup$ – Cham May 31 at 14:26

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