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According to GR, masses (such as the Sun and the Earth) cause space to curve. Then planets such as the Earth follow geodesics on this curved space.

However, if we consider the Earth and the Sun both contributing to the curvature of space, then Earth will be in it's own "dimple" of curved space.

When calculating Earth's orbit, one usually ignores the curvature of space that the Earth creates and just uses the curvature of space that the sun creates (a simple Schwarzschild solution).

What is the justification that we can simply ignore the contribution of a mass such as the Earth (or Mercury) makes to the curvature of space? And that we can simply treat a planet like Earth as a point travelling on a geodesic of space curved only by the Sun?

Since gravity in GR is non-linear, it is far from obvious to me that a planet's own local space-time curvature would not affect it's own orbit in a substantial way. (Obviously this is not the case from experiment). But what is the mathematical justification?

(I am not talking about the gravitational self interaction of a single particle which is another topic in itself but only of very macroscopic objects such as planets).

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  • $\begingroup$ In fact, perhaps a proper treatment of the Earth-Sun system would be something similar to two black holes orbiting one another. (A double Schwarzschild solution???) Then how would one prove that this is approximately gives the same answers as the single Schwarzschild solution with an orbiting point. Minus things like gravitational waves. $\endgroup$
    – user84158
    Commented Feb 15, 2022 at 7:15
  • $\begingroup$ Does a black hole influence its own path through space? $\endgroup$
    – MatterGauge
    Commented Feb 15, 2022 at 7:42
  • $\begingroup$ Even in Newtonian gravity, accurate orbit calculations include the planet's mass. See physics.stackexchange.com/q/3534/123208 $\endgroup$
    – PM 2Ring
    Commented Feb 15, 2022 at 8:23
  • $\begingroup$ I have just taken a look in the volume II of L.Landau, E.Lifchitz : theory of the fields, the part of the equations of a system of cops in second approximation, and I noticed that the potential (φ= k/r=oo .) created by a corp where one wants to calculate the sum of the potential created by others on him is eliminated by a trick called ''normalization'' of the mass, i.e IMPLICITLY, renormalization of infinite curvatures by elimination: elimination of the curvature created by the body itself in its space-time.(elimination of its action on its own geodesic) $\endgroup$
    – The Tiler
    Commented Feb 15, 2022 at 9:21
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    $\begingroup$ @TheTiler "the equations of a system of cops" good one... :) $\endgroup$
    – MatterGauge
    Commented Feb 15, 2022 at 9:29

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I think the answer here by @Nikolaj-K explains it:

In answer to "If the Earth travelling around the Sun is just moving along a straight line in the curved space, shouldn't light also be trapped in orbit around the Sun?"

Just to get that out of the way, "straight line" here means geodesic on a curved surface/manifold, but I guess you understand that.As you've guessed, they don't follow the same geodesic because of their velocity. And the velocity of a body is automatically taken into account, because you don't just compute these geodesics in curved space, but in curved spacetime, where this makes a difference: Imagine a t-x-diagram and how the earth or a ray of light go away from the origin. The earths path will be close to the time axis, while the light path will be leaning towards the space axis (depending on your units). Both paths are similarly affected by the curvature of spacetime but they clearly start out in different directions (in spacetime, not in space) and so the geodesics will be quite different. There is some specific angle for which the object will be orbiting (notice that this now requires at least two spatial dimensions). Smaller angles will fall towards the earth while bigger angles will boldly go where no man has gone before. This is a very geometric notion of escape velocity.

So the geodesic to be calculated the mass of the object following it is within the calculation.

If it were just the sun mass creating the geodesics the complexity of the planetary system would be impossible.

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zooby wrote: "When calculating Earth's orbit, one usually ignores the curvature of space that the Earth creates and just uses the curvature of space that the sun creates (a simple Schwarzschild solution)."

Even under Newton the earth's field affects the trajectory of the sun, which vice versa affects the earth's trajectory. In a proper n-Body simulation you get different results if only the sun affects the earth or if both affect each other, although the difference is not so large in the solar system since the sun's mass is much larger than that of the earth, but it's still there since the earth's mass is not 0. The Schwarzschild solution is only if the mass of the test particles are really neglible compared to the central mass.

zooby wrote: "What is the justification that we can simply ignore the contribution of a mass such as the Earth (or Mercury) makes to the curvature of space?"

If you ignore Mercury's mass it doesn't make much of a difference for the trajectory of the sun, but it's own perihelion shift is more due to the presence of the other planets (532") than the Schwarzschild component (43") of the sun.

The planet's mass doesn't affect its own trajectory in a way that it would accelerate or decelerate in an otherwise flat universe, but it affects the other bodies, which then in turn affect it back differently as they would if they hadn't been affected themselves.

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