Undergrad Solving General Relativity Equations

[Arman777]

Is it really hard to find a solution of a GR equation maybe two planet system ?

Or It could be a different system.I just wonder
1-How much its difficult ( Like can a person calculate those solutions)
2- Whats the boundries (Like we can solve 2 planet system but not 5 etc ? )
3- Can current computers or Quantum Computers can solve them ?
4-Why its so hard to solve them ?

 

[vanhees71]

GR is a non-linear theory, and that makes it hard to find exact solutions, although there are some like the Schwarzschild solution (found in 1916 by Schwarzschild quickly after Einstein had formulated the complete theory), Reisner-Nordstroem solution, Kerr solution, Friedmann-Lemaitre-Robertson-Walker solution.

As far as I know there's no exact self-consistent solution for the two-body problem. Of course you can calculate the motion of a very light body around a very heavy one by taking the heavy body as "source" of the gravitational field, leading to the (outer) Schwarzschild solution and treat the light body as "test body" moving on a geodesic in the fixed Schwarzschild space time. Another way is to use the post-Newtonian approach, i.e., approximate the equations of motion first by the non-relativistic (Newtonian) approximation and then make relativistic corrections in perturbation theory.

 

[PeterDonis]

As far as I know there's no exact self-consistent solution for the two-body problem.

That is my understanding as well. There are numerical simulations but no exact solutions known.

 

[Arman777]

Its so strange...Can they be known in the future ?

 

[PeterDonis]

Can they be known in the future ?

We don't know since we don't know the future.

 

[Arman777]

There are numerical simulations but no exact solutions known.

I didnt quite understand this part..Does it means there are possible many solutions for different numbers and we simulate them ?

 

[vanhees71]

That is my understanding as well. There are numerical simulations but no exact solutions known.

As far as I know, however, only in the post-Newtonian approximation. As in any relativistic classical field theory, classical point particles (or even bodies of finite extension) are very difficult to describe. Already in classical electrodynamics the problem to formulate a consistent dynamical model for classical charged particles is not fully solved (even since Lorentz and Abraham many very ingeneous physicists like Dirac, Feynman, and Schwinger tried it). The best one can do today is the Landau-Lifshitz version of the Abraham-Lorentz-Dirac equation, but that's not fully self-consistent. For GR, the problem is even more complicated.

 

[Ibix]

Peter means that there's no known algebraic solution to two bodies in GR. You can't just add two one-body solutions as you can in Newtonian gravity. However we can use numerical methods to find answers in specific cases.

An example from outside relativity: if you want to know the area under the curve $\sin \left (\theta^2\right)$ then you write $\int\sin\left (\theta^2\right)d\theta$. But no one can do that integral. But you can always plot it on a piece of graph paper and count the squares under the curve and get a numerical approximation.

The same is true in relativity - although you can write down the problem, there are only a handful of cases that are simple enough to solve. The rest we can only count squares on graph paper. (It's more sophisticated than that, but that's the basics of it).

 

[pervect]

Is it really hard to find a solution of a GR equation maybe two planet system ?

Or It could be a different system.I just wonder
1-How much its difficult ( Like can a person calculate those solutions)
2- Whats the boundries (Like we can solve 2 planet system but not 5 etc ? )
3- Can current computers or Quantum Computers can solve them ?
4-Why its so hard to solve them ?

The exact GR equations are a set of non-linear set of partial differential equations. The exact solution to these is not something we can calculate by hand. When you talk about solutions for "2 planets" or "5 planets", you are probably thinking of solving ordinary differential equations, not partial differential equations. Much like electromagnetism, GR is a field theory, so we are solving for the evolution of the field, which fills space-time, not just the position of n planets.

If you think about the existence of electromagnetic and gravitational waves, you may be able to see where the extra complexity comes in. The position of the bodies tells us nothing about the presence, strength, or direction of the waves, so it doesn't give us a complete solution to the problem.

Also note the GR equations are non-linear. This makes the solution of the equations much more difficult.

Its so strange...Can they be known in the future ?

GR does have a formulation as a "well posed initial value problem", which is what I suspect you may be talking about. The features that make a problem well-posed are found in wiki https://en.wikipedia.org/w/index.php?title=Well-posed_problem&oldid=784930135, and some discussion of initial value problems https://en.wikipedia.org/w/index.php?title=Initial_value_problem&oldid=754502948, though the later seems to talk about ordinary differential equations and GR is a set of partial differential equations as previously mentioned. Very very roughly speaking, the basic idea of an initial value problem is that if you know the state of the system in the past, you can evolve the state of the system towards the future - at least as long as there are no singularities. I've skipped over a lot of fine details here, this is a very coarse overview.

One final note. If one is talking about just solving for the motion of planets in the solar system, linearized approximations to GR suffice. These still aren't something a person can carry out by hand. One might be able to run a version of the problem on one's own computer, but for solar system problems it's probably much more practical to run something like JPL's ephemerides, https://ssd.jpl.nasa.gov/?ephemerides.

At the level of accuracy where GR effects are important at all, one needs to include corrections for other effects, such as the shape, or "figure" of the planets, which are neither point masses nor perfect spheres - there's little point to doing GR computations if one ignores other, more important, effects. If you look at the JPL web pages, you can see a list of all the things they take into account.

 

[Arman777]

It was very clear, thanks

 

[TGlad]

I think one reason why there is no two-planet exact solution in GR is because orbiting planets radiate energy in the form of gravitational waves, therefore the solution changes over time, so would need to include a complex collision in the future I suspect. There would be no simple time-symmetric solution.

 

[PeterDonis]

I think one reason why there is no two-planet exact solution in GR is because orbiting planets radiate energy in the form of gravitational waves, therefore the solution changes over time

A solution does not "change over time". A solution is a description of a 4-D spacetime geometry; that description already includes all of the geometry "changes with time" due to gravitational waves.

There would be no simple time-symmetric solution.

Exact solutions don't have to be time symmetric.

 

[PAllen]

As far as I know, however, only in the post-Newtonian approximation. As in any relativistic classical field theory, classical point particles (or even bodies of finite extension) are very difficult to describe. Already in classical electrodynamics the problem to formulate a consistent dynamical model for classical charged particles is not fully solved (even since Lorentz and Abraham many very ingeneous physicists like Dirac, Feynman, and Schwinger tried it). The best one can do today is the Landau-Lifshitz version of the Abraham-Lorentz-Dirac equation, but that's not fully self-consistent. For GR, the problem is even more complicated.

All that aside, numerical relativity is a completely different discipline from post Newtonian approximation. They are sometimes both applied to the same problem as a consistency check. Numerical relativity has been used to model inspiral and merger of neutron stars with various assumed equations of state. It relies on setting up initial data on a Cauchy surface (itself one of the hard parts, because pdiffs must be solved to get consistent initial data), then numerically evolving the EFE.

[edit: for example, here is an introduction to numerical relativity techniques:
https://arxiv.org/abs/gr-qc/0703035 ]

 

[vanhees71]

Indeed, my colleagues in Frankfurt solve full general relativistic hydro, which is a continuum theory, where you don't need the post-Newtonian approximation:

http://astro.uni-frankfurt.de/rezzolla/research/relativistic-hydrodynamicsmhd/

 

[TGlad]

A solution does not "change over time". A solution is a description of a 4-D spacetime geometry; that description already includes all of the geometry "changes with time" due to gravitational waves.
Exact solutions don't have to be time symmetric.

I know they don't have to be time symmetric, but a 2-body problem must spiral inwards due to gravitational waves so any solution must include the full dynamics of a black-hole merger, and the aftermath. It seems much less likely that an exact solution will exist that models all of this.

In fact, I wonder whether the best chance of getting an exact solution is to provide exactly the right amount of cosmological constant to keep the two bodies orbiting without spiraling inwards or outwards. This should lead to a solution which is symmetric to increasing time by one orbit.

 

[PAllen]

I know they don't have to be time symmetric, but a 2-body problem must spiral inwards due to gravitational waves so any solution must include the full dynamics of a black-hole merger, and the aftermath. It seems much less likely that an exact solution will exist that models all of this.

In fact, I wonder whether the best chance of getting an exact solution is to provide exactly the right amount of cosmological constant to keep the two bodies orbiting without spiraling inwards or outwards. This should lead to a solution which is symmetric to increasing time by one orbit.

I'm doubtful cosmological constant would help. Two co orbiting bodies have quadrupole moment that changes direction, and requires that GW be emitted, making an exact solution highly unlikely. A cosmological constant is not going do anything about the changing quadrupole.

 

[TGlad]

I'm doubtful cosmological constant would help. Two co orbiting bodies have quadrupole moment that changes direction, and requires that GW be emitted, making an exact solution highly unlikely. A cosmological constant is not going do anything about the changing quadrupole.

Yes I agree on second thoughts