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Background Info

I'm just starting to look into tensor analysis and general relativity. I have taken college level courses on Newtonian mechanics, electromagnetism, and special relativity that roughly follow the Feynman Lectures on Physics.

This Physics Stack Exchange answer states that

[spacetime] curvature allows and sometimes requires more and/or future curvature, just as a traveling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.

My Question(s)

According to the answer:

what gravitational sources do is allow curvature to react differently to itself than it otherwise would

This raises my first question: How exactly does curvature normally react to itself / evolve without gravitational sources present? And how would we describe this mathematically?

My second question is a followup to the first:

When there are gravitational sources how do they change how curvature reacts to itself? And can this also be described mathematically?

The answer also says that:

curvature allows and sometimes requires more and/or future curvature, just as a traveling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later

My third question is: Is this describing the gravitational wave solution to the Einstein field equations? And how can I mathematically see how the gravitational wave propagates changes in spacetime curvature just like how electromagnetic waves propagate electric and magnetic fields by continuously producing them?

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All this is about solving Einstein equations. These equations are nonlinear equations for the fundamental field, the metric tensor, $g_{\mu\nu}$, describing the geometry of the spacetime. The reaction of curvature to something can be understood as a perturbation of a given solution (to these nonlinear equations). Then, this (small) perturbation propagates on the background of the original solution. Like small water waves on a surface of a flowing river.

  1. Curvature waves (which are nothing else than gravitational waves) are small perturbations which can propagate on the background which can be, for example, a strong curvature of a black hole. Mathematically, the metric $g_{\mu\nu} = \gamma_{\mu\nu} + h_{\mu\nu}$ solves the Einstein equations with $\gamma_{\mu\nu}$ being also a solution of the Einstein equations (without perturbation) and $h_{\mu\nu}$ is a small perturbation, for instance, due to a small change of initial conditions.

  2. By adding matter, via the energy-momentum tensor $T_{\mu\nu}$ to the Einstein equations, we get new solutions $\gamma_{\mu\nu}$ describing the gravitational field of that matter, for example, of a neutron star. Now, gravitational waves $h_{\mu\nu}$, propagating in that background will indirectly feel the presence of the matter (of the star).

  3. The perturbation $h_{\mu\nu}$ of a flat (Minkowski) metric $\gamma_{\mu\nu} = \eta_{\mu\nu}$, describing an empty space, satisfies the wave equation -- identical as the electromagnetic fields do. Consequently, the components of the Riemann tensor, describing the curvature, also satisfy the wave equation. The solutions of the wave equation are gravitational (=curvature) waves.

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  • $\begingroup$ So basically you're saying there is an initial solution to the field equations in a vacuum that evolves a certain way and then in the presence of matter/energy that solution "feels" the presence of the matter and gets perturbed in its background. This doesn't fully explain my first question of how curvature reacting to itself shows up in the field equation though. I heard somewhere that the Schwarzschild metric is a solution that can be thought of as generating its own future curvature so that the solution remains for all time. How would that be explained mathematically? $\endgroup$ – mihirb Jul 22 '20 at 16:13
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    $\begingroup$ See my point 1: Schwarzschild metric is a black hole solution which affects the propagation of gravitational waves. So the (background) curvature affects the curvature waves. $\endgroup$ – Nikodem Jul 22 '20 at 19:34

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