Geodesic Eq Derived from Einstein Field Equations?

[Intrastellar]

TL;DR Summary

Can the geodesic equation be derived from EFEs ?

Since the EFE describes the shape of spacetime, it describes the way black holes, for example, evolve. Can one derive the geodesic equation from it in some limit ?

 

[Ibix]

No. The geodesic equations depend on the connection that you are using. They are a very general tool, applicable to manifolds far more general than just those used in GR.

However, in GR, the connection is uniquely determined by the metric, and thus the specific form of the geodesic equations for a particular spacetime is determined by the metric. And the metric is the solution to the Einstein Field Equations. So once you have solved them for some stress-energy distribution then you have everything you need to write down the geodesics for that spacetime.

So, you can't derive the geodesic equations from the Einstein Field Equations. They are a separate entity. But in GR all the pieces you need to plug into them come from the metric, which is the solution of the Einstein Field Equations.

 

[Intrastellar]

No. The geodesic equations depend on the connection that you are using. They are a very general tool, applicable to manifolds far more general than just those used in GR.

However, in GR, the connection is uniquely determined by the metric, and thus the specific form of the geodesic equations for a particular spacetime is determined by the metric. And the metric is the solution to the Einstein Field Equations. So once you have solved them for some stress-energy distribution then you have everything you need to write down the geodesics for that spacetime.

So, you can't derive the geodesic equations from the Einstein Field Equations. They are a separate entity. But in GR all the pieces you need to plug into them come from the metric, which is the solution of the Einstein Field Equations.

Rather than plugging the metric into the geodesic equation, can you derive for example that two black holes far away from each other will obey an equation with a second order derivative in proper distance and 2 first order derivatives contracted with the connection in terms of the metric?

 

[PeterDonis]

can you derive for example that two black holes far away from each other will obey an equation

No. There is no known exact solution for a spacetime with more than one black hole (or more than one gravitating mass of any type, for that matter). All such cases can only be analyzed numerically.

 

[PeterDonis]

Rather than plugging the metric into the geodesic equation, can you derive for example that two black holes far away from each other will obey an equation with a second order derivative in proper distance and 2 first order derivatives contracted with the connection in terms of the metric?

What you are describing is not something different from "plugging the metric into the geodesic equation". It is "plugging the metric into the geodesic equation".

 

[Intrastellar]

No. There is no known exact solution for a spacetime with more than one black hole (or more than one gravitating mass of any type, for that matter). All such cases can only be analyzed numerically.

What about as an approximation ? I assume that to derive something that looks like the geodesic equation you have to take a number of approximations.

 

[PeterDonis]

I assume that to derive something that looks like the geodesic equation you have to take a number of approximations.

You assume incorrectly. The geodesic equation can be derived without any approximations whatever.

 

[Intrastellar]

You assume incorrectly. The geodesic equation can be derived without any approximations whatever.

Nononono I mean derived from the EFE as an approximation.

 

[PeterDonis]

I mean derived from the EFE as an approximation.

No. "Approximation" is not a magic word that somehow makes what @Ibix said previously invalid.

 

[Intrastellar]

What I have in mind is you take the equations obeyed by two far away black holes and approximate them to get some notion of trajectory and coordinates of these two objects, and then derive that they will approach each other following a particular trajectory using EFEs, then that would consititute a derivation of the geodesic equation from the EFE.

Nothing that I said contradicts what ibex said in any way

 

[Intrastellar]

Perhaps a better question is : if you solve EFEs numerically for far away black holes, will the black holes approximately satisfy the geodesic equation in their approach ?

 

[PeterDonis]

Perhaps a better question is : if you solve EFEs numerically for far away black holes, will the black holes approximately satisfy the geodesic equation in their approach ?

Yes, this is a much better question since it describes a reasonable approach.

To clarify the approach, when you say "satisfy the geodesic equation", this would have to mean that the trajectories of the black holes would be (approximately) geodesics of the overall spacetime geometry, as computed using the connection derived from the metric of that spacetime geometry.

There is also a technical point here: black holes aren't ordinary objects like planets or stars, and the regions inside their horizons have many counterintuitive properties, including not having "centers of mass" the way ordinary objects do. That means a black hole, strictly speaking, doesn't have a "trajectory" the way an ordinary object does. We can finesse this point by considering a "world tube" whose surface is outside the horizons of the holes, and computing what the trajectory of the center of mass would be if there were an ordinary object instead of a black hole inside the tube.

With those technical points given, AFAIK the answer to the question is yes.

 

[Intrastellar]

Great, thanks! Now my remaining question is: is it possible to reflect this fact analytically ?

 

[PeterDonis]

is it possible to reflect this fact analytically ?

Not if no analytical solution is known for the scenario in question. Go read post #4 again.

 

[Intrastellar]

Not if no analytical solution is known for the scenario in question. Go read post #4 again.

Do you need an analytical solution to check such a property of trajectories? Surely the question is simpler than to require the full GR simulation of two black holes.

 

[PeterDonis]

Do you need an analytical solution to check such a property of trajectories?

You need an analytical solution if you want to "reflect" anything analytically, which is what you asked.

 

[PeterDonis]

Surely the question is simpler than to require the full GR simulation of two black holes.

Nothing less than a "full simulation" will give you the kind of information you would get from an analytical solution if you had one.

 

[Intrastellar]

You need an analytical solution if you want to "reflect" anything analytically, which is what you asked.

Perhaps I am using the word analytically incorrectly. What I meant is symbolically or mathematically, through algebra and calculus, and by taking approximations where necessary.

 

[PeterDonis]

What I meant is symbolically or mathematically, through algebra and calculus, and by taking approximations where necessary.

Yes, that's what "analytically" means, and that's the meaning I was using in my previous answers.

 

[Intrastellar]

Not if no analytical solution is known for the scenario in question. Go read post #4 again.

How can you say that there is no analytic solutions when you allow approximation as part of the definition of analytical solution ? There is, in the approximation that one of them has zero mass!

Anyway, my question is more that is there a general argument that one can make for why such a result is to be expected from the EFEs alone.

 

[PeterDonis]

How can you say that there is no analytic solutions when you allow approximation as part of the definition of analytical solution ?

Approximation is not the same as numerical simulation. You can't do an analytical approximation if there is no exact analytical solution to approximate from. Approximation can make the equations simpler, but it can't conjure up equations from nowhere.

There is, in the approximation that one of them has zero mass!

That's not an approximation. It's a different problem, to which an exact analytical solution happens to be known (the solution for a single gravitating mass, or more accurately the family of such solutions parameterized by various properties of the gravitating mass).

 

[PeterDonis]

my question is more that is there a general argument that one can make for why such a result is to be expected from the EFEs alone.

Do you mean the result that gravitating masses are expected to follow (at least approximately) geodesics of the overall spacetime geometry to which they contribute?

 

[Intrastellar]

Do you mean the result that gravitating masses are expected to follow (at least approximately) geodesics of the overall spacetime geometry to which they contribute?

Yes, except that I want to reach this conclusion without assuming the geodesic equation, starting just from the EFES.

 

[PeterDonis]

I want to reach this conclusion without assuming the geodesic equation, starting just from the EFES.

You can't. Without the geodesic equation, you don't know which trajectories are geodesics.

This thread is going around in circles.

 

[Intrastellar]

You can't. Without the geodesic equation, you don't know which trajectories are geodesics.

This thread is going around in circles.

You don't need to know, you derive the trajectories from the EFE, and compare them with those of the geodesic equation.

 

[PeterDonis]

you derive the trajectories from the EFE, and compare them with those of the geodesic equation.

Which you can't do if you don't know the geodesic equation.

Since the thread is continuing to go around in circles, it is now closed.