Modeling the Earth and Sun (2 body orbits) using general relativity?

[James1238765]

TL;DR Summary

How to model earth and sun (2 body orbits) using general relativity?

Modeling the time evolution of the sun and earth orbiting each other using $F = \frac{GMm}{r^2}$ is straightforward. However, it appears that modeling the time evolution of the same 2 body system using general relativity seems to be a hard/intractable problem?

There was in depth discussion by @bcrowell in https://www.physicsforums.com/threads/multibody-system-in-general-relativity.362949/ regarding multibody modeling using general relativity.

As with the OP in the linked thread, I am not so much interested in extreme conditions like relativistic speed, time dilation, or black holes. I would only like to model a simple 2 body problem with the sun and the earth orbiting each other using general relativity.

Any experience or reference to resources for doing this is much appreciated.

 

[phyzguy]

This is a very interesting problem. For the Kepler orbit (Newtonian, no GR), it is possible to re-write the equations of motion to read: $$\frac{d^2u}{d\phi^2} + u = \frac{GM}{h^2}$$ where u = 1/r where r is the radius, and h is the specific angular momentum, which is a constant of the motion. You can find this in Goldstein "Classical Mechanics" for example, or at this Wikipedia page.
It turns out that with the sun/Earth, GR just introduces a small correction, so the equation becomes: $$\frac{d^2u}{d\phi^2} + u = \frac{GM}{h^2} + \frac{3 GM}{c^2} u^2$$
You can try solving this numerically, using Runge-Kutta methods, for example. You should be able to derive the precession of the perihelion of Mercury due to GR in this way.

 

[James1238765]

@phyzguy thank you. I will post a simulation result of the differential equation with GR correction, as I happen to have a Runge Kutta implementation.

I wonder, from

$$R_{ij} - \frac{1}{2}Rg_{ij} = T_{ij}$$

it seems that, two bodies moving will just change the $\Gamma$ numbers around the space they are located at, like a person taking consecutive steps on the bed will bend the bed surface according to their footsteps...

If one numerically superimposes two Schwarzchild $\Gamma$ numbers all over a 4D grid (details to be worked out), one set of numbers representing the sun and the other set of numbers representing the earth, does this allow for a test particle like the moon to correctly follow an orbit influenced by both the sun and the earth?

 

[Ibix]

does this allow for a test particle like the moon to correctly follow an orbit influenced by both the sun and the earth?

No. The Einstein Field Equations are not linear so you can't superpose solutions, which is one part of why gravity is so difficult. If you simply superpose two Schwarzschild metrics and turn the crank you'll find non-zero stress energy everywhere. It won't be much for the Earth-Moon system, because that doesn't deviate much from Newtonian, but it'll be more severe for more extreme circumstances.

Detailed numerical simulations of the two body problem in relativity are primarily the domain of people with access to large clusters, is my understanding.

 

[James1238765]

@Ibix thank you. What are the basic methods to simulate two body problems, as in what do they input into the initial $\Gamma$ data of the system? Is there a known $g$ metric solution for a 2 body system?

 

[Ibix]

There is no known solution for a two-body problem.

I believe the general approach is that you start with a reasonable guess for a spatial metric on a Cauchy surface and iterate towards something that satisfies the field equations on that surface given your stress-energy distribution. Then you propagate that forward. I've never done it.

 

[phyzguy]

Yes, in general it's very difficult. You need to simultaneously solve for the metric and the motions of the bodies in a way which is self consistent. People worked for decades and were unable to get the codes to converge. Then in the early 2000's they evolved new techniques which allowed the codes to converge. This allowed the simulations of merging black holes that we see today. You could start here.

 

[James1238765]

@Ibix. thank you, it is something to start with.

To wax philosophical for a bit, nature has to "solve" this $\Gamma$ numbers too for it to function. Suppose two bodies A and B are stationary at their respective location. All the $\Gamma$ numbers correctly represent the $T_{ij}$ of this universe.

Then, B takes an infinitesimal step in some direction. If this means that the $\Gamma$ numbers will suddenly change like crazy all over the universe, would it not raise questions if this adjustment process is can be achieved entirely locally?

How much is known about how the nonlinear changes to $\Gamma$ occur, when a small change in position is made to a previously "solved" system?

 

[FactChecker]

As a practical matter, this sounds like something where the difference between Newtonian and GR solutions is tiny compared to the basic, Newtonian solution (not to mention the effect of massive bodies like Jupiter and Saturn). Getting a correct iterative solution would require extreme measures to reduce round-off errors. It might be more practical and accurate to estimate the difference (GR-Newtonian) and use that to adjust the Newtonian solution.
(PS. I realize that I am way out of my lane here and will leave this to others who are experts.)

 

[James1238765]

@FactChecker personally I am drawn to GR mainly because the construction is clear. The solution might be impossibly hard and beyond us, which is fine, but the problem statement is clear. Unless we develop FTL ships or are going into blackholes, practically yes Newton gravity should be plenty sufficient for daily uses...

 

[PeterDonis]

it appears that modeling the time evolution of the same 2 body system using general relativity seems to be a hard/intractable problem?

No exact solution is known for the two-body problem in GR (unlike Newtonian physics).

Under certain conditions, such as weak fields and slow motion, you can come up with good approximate solutions using the Einstein-Infeld-Hoffman equations and numerically integrating them. Nowadays with computers doing that is tedious, but not intractable.

https://en.wikipedia.org/wiki/Einstein–Infeld–Hoffmann_equations

 

[Ibix]

Then, B takes an infinitesimal step in some direction. If this means that the $\Gamma$ numbers will suddenly change like crazy all over the universe, would it not raise questions if this adjustment process is can be achieved entirely locally?

They don't suddenly change, and changes propagate within future light cones, but they do change everywhere all the time (at least in principle). The orbiting bodies emit gravitational radiation that radiates to infinity, and is therefore everywhere changing all the time. In an Earth-Moon simulation this is entirely negligible, but in an extreme circumstance like black holes the evolution of the waves and their interactions (non-linear, remember, they don't simply pass through each other) are very important to the orbits.

Additionally, if you aren't considering just black holes, you need equations of state describing how the stress-energy evolves as a result of its current state and changes to space around it. Those go into the stress-energy tensor and need to be solved as part of the EFEs.

It all boils down to a lot of computer power.

 

[James1238765]

thank you. It seems that time evolution methods like post-Newtonian starts with the Newton attraction and adds some correction terms to account for GR...

From https://www.physicsforums.com/threads/multibody-system-in-general-relativity.362949/ , it is implied that even if all the $\Gamma^{i}_{jk}$ and $T_{ij}$ have been solved corrected for a 2-body system, this structure will only affect other test particles' trajectories, and not each other(!), so the two bodies A and B still will not affect each other gravitationally purely through general relativity (It is said that body B will fly off into far space just like a test particle, instead of mutually attracting with body A like Newton said).

This seems to be a problem, as how can a theory of gravity not give rise implicitly to gravitational attraction between 2 bodies (as Newton's gravity does)?

 

[James1238765]

In any case, the first step has to be to find (by any means) the correct $\Gamma$ numbers for a stationary 2-body system like below:

so that $T_{ij}$ = 0 correctly for all points outside the two masses.

How can we find the correct $\Gamma$ numbers that satisfy this system? To search and test exhaustively different 64 values (0.01, 0.02, 0.03...) of the $\Gamma$ numbers on each location would be prohibitively expensive by far.

 

[PeterDonis]

a stationary 2-body system like below:

There is no such thing as a stationary 2-body system in the presence of gravity, if by "stationary" you mean that the two bodies don't move.

 

[James1238765]

Technically I meant what strategies could we use to mathematically find the correct $\Gamma$ numbers for a 2 body situation like:

such that $T_{ij}$ = 0 for all points outside the bodies?

 

[PeterDonis]

Technically I meant what strategies could we use to mathematically find the correct $\Gamma$ numbers for a 2 body situation like:

View attachment 320983

such that $T_{ij}$ = 0 for all points outside the bodies?

I'm not sure why you're focusing on the $\Gamma$ numbers. What you want is a metric, but, as already noted, there is no known exact solution for a metric for a two body system in GR. Such cases are solved numerically in practice (for example, to calculate the expected gravitational wave emission and changes in orbital parameters for binary pulsars).

For a case like the solar system, nobody bothers trying to solve the Einstein Field Equation numerically because there's no need to; instead equations like Einstein-Infeld-Hoffmann are used, which are basically the Newtonian equations for a system of many point masses with GR correction terms added. For many systems, like the solar system, only the first few correction terms are needed (IIRC going to order $v^2 / c^2$ is good enough for almost any solar system calculation, although some very precise experiments have probed out to, IIRC, order $v^5 / c^5$).

 

[James1238765]

I'm not sure why you're focusing on the $\Gamma$ numbers

thank you. because the 64 $\Gamma^i_{jk}$ numbers by themselves complete specify everything on the left hand side of the GR field equation, ie. $R^i_{jkl}, R_{ij}, R$ are all derived quantities from $\Gamma$. The metric $g_{ij}$ itself need not have a nice analytical description, as long as the prescription is clear for what values of $\Gamma$ to put at every location.

 

[PeterDonis]

the 64 $\Gamma^i_{jk}$ numbers by themselves complete specify everything on the left hand side of the GR field equation, ie. $R^i_{jkl}, R_{ij}, R$ are all derived quantities from $\Gamma$.

You can say the same about the 10 numbers that are the independent components of the metric: everything in the field equation is derived from them (their first and second derivatives).

Also, there aren't 64 independent $\Gamma$s, because they are symmetric in the two lower indexes. There are only 40, and that's without specifying any coordinate choices to further constrain them.

 

[Ibix]

From https://www.physicsforums.com/threads/multibody-system-in-general-relativity.362949/ , it is implied that even if all the $\Gamma^{i}_{jk}$ and $T_{ij}$ have been solved corrected for a 2-body system, this structure will only affect other test particles' trajectories, and not each other(!), so the two bodies A and B still will not affect each other gravitationally purely through general relativity (It is said that body B will fly off into far space just like a test particle, instead of mutually attracting with body A like Newton said).

No, @bcrowell said that a linearised version of GR has that problem, citing MTW p186 if anyone wants to look that up.

 

[James1238765]

@Ibix even in the full-mode GR, there is no state-evolution equation in the style of Schrodinger equation, is there? For instance, if the state of a two body currently is:

which equation in GR gives us the next state-evolution of this system given the current state?

 

[vanhees71]

There is no known solution for a two-body problem.

I believe the general approach is that you start with a reasonable guess for a spatial metric on a Cauchy surface and iterate towards something that satisfies the field equations on that surface given your stress-energy distribution. Then you propagate that forward. I've never done it.

I think a standard strategy is to use PPNP (parametrized post-newtonian parametrizations), i.e., you look at the Newtonian two-body problem, which can be solved in the usual way by describing it as uniform center-mass motion and the relative motion, which is just the motion of a single "quasiparticle" with the reduced mass $\mu=m_1 m_2/(m_1+m_2)$ in the Newtonian gravitational potential $-G m_1 m_2/r=-G \mu M/r$ and then doing systematic relativistic corrections.

For the planets in our solar system, it's of course accurate enough to describe the planet as a "test particle" moving in the spacetime due to the presence of the Sun, i.e., in first approximation in the Schwarzschild space describing the Sun as a radially symmetric mass distribution.

 

[Ibix]

which equation in GR gives us the next state-evolution of this system given the current state?

That's not really how GR works. A solution to the field equations is 4d, so it's a complete history of everything in the universe. There isn't naturally a notion of "now" in order to have a state "now".

That said, you can consider slicing spacetime into a stack of "space now" slices and consider the geometry of one slice and the relationship between the geometry of one slice and the next. That involved limiting yourself to spacetimes where that makes sense (e.g. no closed timelike curves, since the behaviour of a slice depends on itself), but that isn't unreasonable. Do note, though, that its success depends on making a reasonable guess as to what a sensible initial "now" is, and that there's a complex problem in solving for what the geometry on that slice is before you start propagating. The relevant thing to search for there is the ADM formalism, as well as stuff on relativity as an initial value problem. Wald has a chapter on the latter, at least.

 

[James1238765]

@Ibix In the Schwarzchild metric, Kerr metric and most others, time seems to be a stepchild which seems to be sitting there just to fit in with the established spacetime idea in special relativity? $t$ does not influence or appear anythere in these metrics...

To encode time evolution into this 4D idea of space-time, it seems that one essentially needs to first solve the entire effectively 3-dimensional $\Gamma^{i}_{jk}$ and $T_{ij}$ numbers everywhere in space for 1,000,000 time frames into the future, then list out the whole set of numbers for 1,000,000 time frames into the future, and then one can say that this 4D spacetime now "contains" the time evolution encoded into it?

 

[PeterDonis]

In the Schwarzchild metric, Kerr metric and most others, time seems to be a stepchild which seems to be sitting there just to fit in with the established spacetime idea in special relativity?

No.

$t$ does not influence or appear anythere in these metrics...

These spacetimes are stationary spacetimes. That means one can choose coordinates in which none of the metric coefficients are functions of $t$; those are the coordinates you are using to seeing them in. It doesn't mean there is no time in these spacetimes or that time has no meaning in them.

Not all spacetimes are stationary.

To encode time evolution into this 4D idea of space-time

In a stationary spacetime, "time evolution" just means the geometry stays the same. You don't need to "encode" anything; it's already there in the solution.

it seems that one essentially needs to first solve the entire effectively 3-dimensional and numbers everywhere in space for 1,000,000 time frames into the future, then list out the whole set of numbers for 1,000,000 time frames into the future, and then one can say that this 4D spacetime now "contains" the time evolution encoded into it?

No. This is all nonsense. Time is already there in the solutions. You don't need to add anything. The fact that the metric is not a function of $t$ does not mean the solution is "effectively 3-dimensional" or that you need to add something to it to have time in it.

 

[James1238765]

The reason nothing time evolves in the Schwarzchild metric is due to all the $\Gamma$ numbers not changing over the different $t$ coordinates.

 

[PeterDonis]

The reason nothing time evolves in the Schwarzchild metric is due to all the $\Gamma$ numbers not changing over the different $t$ coordinates.

No, the reason nothing changes with time in Schwarzschild spacetime is that the spacetime is stationary. The coordinate property you describe is just a side effect of that plus a particular choice of coordinates.

But it's wrong to say that the fact that nothing changes means that nothing time evolves. All "nothing changes" means is that that is what "time evolution" is in a stationary spacetime. It doesn't mean it's not a spacetime or doesn't include time.

 

[vanhees71]

In special relativity the spacetime-metric components are even just constant in a (pseudo-)Euclidean basis, i.e., $g_{\mu \nu}=\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$. What you say that in SRT nothing time evolves? Why should anybody have come to the idea to use Minkowski space to describe the motion of charges in electromagnetic fields or of the electromagnetic fields themselves? Or would you conclude from the fact that the metric components do not depend on the spatial coordinates in this choice of Lorentzian coordinates that it cannot describe changes in space?

In short: I think you have a fundamental misunderstanding about the meaning of the "metric" already in SRT. So perhaps it's a good idea to look at SRT again before studying GRT.

 

[James1238765]

@vanhees71 I find that it's hard to always discuss physics conceptually here. What I meant was, what happens when two stars are close to one another is that they will pull each other closer over time.

But nothing moves in the Schwarzchild metric. Inserting a $t$ velocity with $\Delta s$, the test particle will move, but the bodies generating the curvature itself will not move. Why is this? Because the metric is not dependent on time. A more enlightened metric that changes the $\Gamma$ values over time might be able to encode this gravitational pull between two objects.

But because the solutions we have such as Schwarzchild are not $t$ dependent, this is why the picture is frozen. Thus we need to resort to other methods like Newton's gravity to "update" the positions outside of GR. One may endlessly discuss conceptually whether time is really a dimension or a consecutive rendering of frames, but I would rather calculate and not go into this.

 

[James1238765]

Let's try this calculation instead. The following is a rendering of $T_{ij}$ of a Schwarzchild metric:

Most of the values vanish except at a pointlike source at the center. I would now like to superpose two of these point sources, as follows:

Without prejudgement what the resulting $T_{ij}$ might look like, let's find the technical steps to do this.

$R_{ij}$ is completely dependent on $\Gamma^i_{jk}$, but $R$ depends also on $g^{ij}$ which in turn depends on $g_{ij}$.

Suppose there are two source metrics $g1_{ij}$ and $g2_{ij}$ separated some distance apart as in the picture. What should be added to try to obtain the second picture, termwise $g1_{ij} + g2_{ij}$? How should the distance apart be represented, since $g_{ij}$ is written in terms of $\{r,\theta,\phi\}$ coordinates from each's own origin?

 

[PeterDonis]

what happens when two stars are close to one another is that they will pull each other closer over time

Yes.

nothing moves in the Schwarzchild metric

That's because the Schwarzschild metric doesn't describe a spacetime with more than one gravitating mass in it. It only describes the vacuum region outside a single gravitating mass, with nothing else present (except test objects that don't affect the spacetime geometry).

A more enlightened metric that changes the $\Gamma$ values over time might be able to encode this gravitational pull between two objects.

As you have already been told multiple times in this thread, there is no known exact solution for a metric for multiple gravitating bodies. You have to solve that case numerically, as is done, for example, for binary pulsars.

Thus we need to resort to other methods like Newton's gravity to "update" the positions outside of GR.

No, that's not what we are doing when we use Newtonian gravity, or a generalization of it that includes GR corrections, like the Einstein-Infeld-Hoffmann equations.

What we are doing when we use those tools is to use approximations to GR that give good enough answers for the particular problems for which we use them, and which are much easier than solving numerically the full Einstein Field Equation. None of this is "outside of GR".

would now like to superpose two of these point sources

What you are doing here is a linear approximation of GR, which is one option for numerical solution (generally speaking the equations won't have exact analytical solutions). Superposing multiple sources is equivalent to assuming that the nonlinearities in the full Einstein Field Equation are too small to matter for the particular problem you are addressing.

How should the distance apart be represented

Obviously you have to adopt a single coordinate chart for the entire problem, and figure out how to transform the two metrics you are superposing into the new chart. It might be easier conceptually if you use the Cartesian form for the metric due to each source, since then you can just pick a coordinate chart whose spatial origin is, say, the center of mass of the two-body system, and the metric due to each source is then just a simple translation to put it in the new coordinates.

 

[James1238765]

"the Cartesian form for the metric due to each source, since then you can just pick a coordinate chart whose spatial origin is, say, the center of mass of the two-body system, and the metric due to each source is then just a simple translation to put it in the new coordinates."

I am afraid you will need to be more specific than words. What are the transformation equations?

 

[James1238765]

Is it possible to recover $g_{ij}$ completely from all the descendant $\Gamma^i_{jk}$?

Adding $\Gamma^i_{jk}$ can easily be done gridpoint-wise. The main problem is the combined metric $g^{ij}$ is needed to calculate R. This combined metric is probably ugly looking, but can it be integrated fully from its descendant $\Gamma^i_{jk}$?

The formula for a single $\Gamma$ is:

$$\Gamma^a_{bc} = \frac{1}{2}g^{aa}(\frac{dg_{aa}}{dc} + \frac{dg_{ca}}{db} - \frac{dg_{bc}}{da}) + \frac{1}{2}g^{ab}(\frac{dg_{ab}}{dc} + \frac{dg_{cb}}{db} - \frac{dg_{bc}}{db}) $$
$$+ \frac{1}{2}g^{ac}(\frac{dg_{ac}}{dc} + \frac{dg_{cc}}{db} - \frac{dg_{bc}}{dc}) + \frac{1}{2}g^{ad}(\frac{dg_{ad}}{dc} + \frac{dg_{cd}}{db} - \frac{dg_{bc}}{dd})$$

Is no information lost going from $g_{ij}$ to the full set of $\Gamma^i_{jk}$ and vice versa?

 

[PeterDonis]

What are the transformation equations?

If you have the metric for each individual gravitating mass expressed in Cartesian coordinates, then in center of mass Cartesian coordinates the transformation is a spatial translation (a different one for each mass). The translation amount for each mass will be a function of coordinate time, which is where the time dependence due to the mutual gravity of the masses will appear.

For an appropriate choice of coordinates, you should be able to make, say, the $x$ axis be along the line between the two masses, in which case the spatial translation for each object would look something like $x' = x - r(t')$, where $r(t')$ is the distance of the object from the system center of mass as a function of coordinate time in the new coordinates. The disadvantage of doing this is that, in the general case (i.e., if you are not restricting to the masses themselves moving only radially, so that they just come together, bounce off each other, and recede, possibly in an unending cycle), test objects at infinity, such as a distant reference star, will not be at rest in the coordinates.

 

[Ibix]

@Ibix In the Schwarzchild metric, Kerr metric and most others, time seems to be a stepchild which seems to be sitting there just to fit in with the established spacetime idea in special relativity? does not influence or appear anythere in these metrics...

These are metrics one obtains by asserting that there is a sense in which nothing changes over time. Naturally the metrics one obtains do not depend on time. But this is not a valid assumption for, e.g., orbiting planets, and the metrics you get will inevitably depend on time (probably on all four coordinates, actually, since I don't think you have any symmetries except a mirror symmetry in the ecliptic plane).

A more enlightened metric that changes the $\Gamma$ values over time might be able to encode this gravitational pull between two objects.

Of course. But there is no known analytical solution to this case, so you need to solve the problem numerically.

I am afraid you will need to be more specific than words. What are the transformation equations?

You write the Jacobian matrix $J_i{}^j=\partial x'^i/\partial x^j$ where the $x'$s are the new $t,x,y,z$ coordinates and the $x$ are the standard $t,r,\theta,\phi$ coordinates (and checking that I got the partial the right way up). Then you compute the metric in the new coordinates $g'_{ij}=J_i{}^kJ_j{}^lg_{kl}$. Then you can just offset your two masses by some $\pm z_0$ and add the two metrics. Crank out the Einstein tensor and you'll find it's non-zero everywhere. That's why you can't add metrics - but one of the spacelike planes might be a starting point for a correct solution.