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All,

The recipe for classical mechanics for one particle: come up with expressions for all forces, solve $\,F = ma\ $ to arrive at position as a function of time.

The recipe for quantum mechanics: come up with an expression for the potential energy, solve Schrödinger's equation to arrive at the wavefunction.

(Of course, the devil is in the details, and often analytic solutions are not possible, so approximations or numerical methods must be resorted to, but that is not what I am talking about here.)

Is there a “recipe” for general relativity? (I have in mind here, solving for one particle.) I want to say that this is the recipe:

  1. Use the mass/energy distribution to come up with a stress tensor.
  2. Use the stress tensor in Einstein's field equations to solve for the metric.
  3. Use the metric in the geodesic equation to solve for the particle's coordinates as a function of time.

Again, I'm sure these steps can't be done analytically in most cases, I just want to know if the idea behind my "recipe" is correct.

Also, I've noticed that the Schwartzschild metric explains many of the interesting results of GR: perehelion advance, GPS time correction, bending of light, and some properties of black holes. How often is this metric used by real-life GR researchers?

In real-life GR research, do physicists derive metrics? Or do they start with a reasonable metric (maybe from symmetry arguments) and go from there? Probably both, but is it $50/50$, or usually one way, not the other?

Thanks,

B.W.

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There are various research topics one can pursue in GR. Starting with a given metric as a background and analysing particle's / field's dynamics often yields interesting results. This approach is not accurate, though. The heavier the particles / fields (E=mc^2 so 'heavier' has to be interpreted accordingly) the less accurate the model would be.

It is perfectly fine to use the Schwarzschild metric to derive Mercury perihelion advance or time differences between the Earth surface and the orbit because in both cases there is one single spherical massive object generating the spacetime distortion in those cases - namely the Sun and the Earth. However, should Mercury become a black hole of a similar diameter, that computation would be useless. Physics is an art of appropriate approximation.

If the fixed metric approximation is not accurate enough we have to resort to the 'full' approach:

  1. Einstein equations that connect the metric with Stress-Energy of the system
  2. Equations of motion for the system that take the spacetime curvature into account
  3. Restriction to certain symmetries (like spherical or radial) can be helpful to simplify the system at the expense of restricted degrees of freedom for your field/particles
  4. Gauge freedom has to be set (e.g. lapse and shift in the ADM formulation)
  5. Boundary values and initial values for the resulting system of PDEs

Once you have those they uniquely define the dynamics. You need to solve the PDE's which you often do numerically - that is a big part of todays GR.

To sum up, the 'real' GR research tries to crack the full equations and this is what people always try to do. Only if you find those equations too difficult to approach you resort to fixed metrics - unless the task makes it obvious that non-trivial spacetime dynamics is not relevant - as in the cases mentioned in the question.

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