What's the underlying frame of the Einstein's Field Equation?

[Pyter]

TL;DR Summary

In what coordinate system is set up and solved the Einstein's Field Equation and how does it relate to our continuum?

Hello all,

I have a question on a pivotal concept of GR that I've never managed to fully grasp.

In what coordinate system is the Einstein's Field Equation set up and solved?

I've always assumed it's an Euclidean 4D space, whose metric is irrelevant because we are dealing with scalar functions, solutions of the ten independent differential equations.
But if it's so, how does it map to our 4D curved continuum? I've always seen our curved spacetime only mapped to a "local" flat space in the infinitesimal region around a given point, but never "globally".

What if a want to solve the EFE in an extended spacetime region, i.e. our solar system for several days?

From a procedural standpoint, should I have to take a "snapshot" of the masses in my curved spacetime to express the stress-energy tensor as a function of my curvilinear coordinates, map it to a 4D flat space, solve the EFE to obtain the metric tensor g as a function of Euclidean coordinates, then map it back to my curved spacetime?

What boundary conditions should I use i.e. for velocities, expressed in which coordinate system?

And what would be the use for such a g? To compute the motion equations of the masses in my curved spacetime?

 

[vanhees71]

GR is generally covariant, i.e., the Einstein(-Hilbert) field equations look the same in all coordinates (reference frames). If it comes to the solution of these non-linear (and thus complicated) equations of motion, of course a clever choice of coordinates is crucial. All the exact solutions (like the Schwarzschild, Kerr, and FLRW solutions) of course result from having many symmetries and the choice of coordinates is according to these symmetries.

 

[Dale]

Summary:: In what coordinate system is set up and solved the Einstein's Field Equation and how does it relate to our continuum?

In what coordinate system is the Einstein's Field Equation set up and solved?

You can use any coordinate system. It is valid in all of them.

Summary:: In what coordinate system is set up and solved the Einstein's Field Equation and how does it relate to our continuum?

What if a want to solve the EFE in an extended spacetime region, i.e. our solar system for several days?

For something like that you are probably going to want to use the ADM formalism:

https://en.m.wikipedia.org/wiki/ADM_formalism

This is the usual approach for solving the EFE in terms of boundary conditions and so forth.

 

[vanhees71]

In addition you need such a (1+3)-formalism to formulate the corresponding initial-value problems and solve them numerically. For the application in relativistic transport/hydro simulations to predict gravitational-wave signals, see

L. Rezzolla and O. Zanotti, Relativistic hydrodynamics,
Oxford University Press, Oxford (2013),
https://dx.doi.org/10.1093/acprof:oso/9780198528906.001.0001

 

[PeterDonis]

I've always assumed it's an Euclidean 4D space

Spacetime is Lorentzian, not Euclidean.

whose metric is irrelevant because we are dealing with scalar functions, solutions of the ten independent differential equations.

Those ten solutions are the metric (each of the ten functions is one of the ten independent components of the metric, and the ten independent differential equations are differential equations involving various combinations of those functions), so the metric is not irrelevant.

 

[Pyter]

So any frame is good. But since I observe the stress-energy tensor in my curvilinear frame, am I forced to use that frame? Or should I come up with any arbitrary map that simplifies the EFE, solve it in that frame, then map it back to my lab frame?

And once I have the metric tensor, I solve the equations of motion of my masses by imposing that they follow geodesics? The gravity is not thus considered a force, but it only warps the spacetime through the metric?

 

[Pyter]

Spacetime is Lorentzian, not Euclidean.Those ten solutions are the metric (each of the ten functions is one of the ten independent components of the metric, and the ten independent differential equations are differential equations involving various combinations of those functions), so the metric is not irrelevant.

I meant that I don't need the metric of the 4D flat (Euclidean, Minkowski or whatnot) space I assumed the EFE had to be solved into, since I'm not using vectors. I know that the solution is the metric of my warped continuum.

 

[PeterDonis]

I observe the stress-energy tensor in my curvilinear frame

You make actual measurements of tensor components using particular measuring devices, which can be used to define a particular frame, but you can transform those measured components into any frame you like for the purpose of making calculations, including the calculations required to solve the EFE.

once I have the metric tensor, I solve the equations of motion of my masses by imposing that they follow geodesics?

Assuming that there are no forces present, yes. But that won't always be the case.

gravity is not thus considered a force, but it only warps the spacetime through the metric?

In GR, yes, that is correct.

 

[PeterDonis]

I meant that I don't need the metric of the 4D flat (Euclidean, Minkowski or whatnot) space I assumed the EFE had to be solved into

You assumed wrong. You solve the EFE for your 4D spacetime; the solution gives you the metric of that spacetime. That metric won't be flat unless your stress-energy tensor vanishes everywhere, and even then flat Minkowski spacetime is not the only possible vacuum solution.

I'm not using vectors.

How are you not using vectors? Tensors are vectors--more precisely, what you are calling "vectors" are just 1st-rank contravariant tensors, but tensors of any rank form a vector space.

 

[Pyter]

You assumed wrong

I'm aware of that now, thank you all.

How are you not using vectors?

In the sense that to solve the differential scalar equations I don't need them. I'll use the ds² later when I set up the geodesics equations with the solved g.

 

[Dale]

Or should I come up with any arbitrary map that simplifies the EFE, solve it in that frame, then map it back to my lab frame?

That is usually the best approach. If there is some symmetry, then using coordinates with that same symmetry is often helpful

 

[Pyter]

That metric won't be flat unless your stress-energy tensor vanishes everywhere, and even then flat Minkowski spacetime is not the only possible vacuum solution.

That's interesting, maybe topic for another thread. I haven't delved into it so far, but I figured that the homogeneous EFE only admitted the identically flat space solution or the planar wave solution, for analogy with the homogeneous Maxwell equations.

 

[PeterDonis]

I figured that the homogeneous EFE only admitted the identically flat space solution

As you have seen, that is not correct. Other important vacuum solutions besides flat Minkowski spacetime include Schwarzschild spacetime and Kerr spacetime.

for analogy with the homogeneous Maxwell equations.

The key difference between them is that Maxwell's equations are linear, while the EFE is nonlinear.

 

[Dale]

Other important vacuum solutions besides flat Minkowski spacetime include Schwarzschild spacetime and Kerr spacetime.

And of course most of the typical gravitational wave spacetimes.

 

[PeterDonis]

And of course most of the typical gravitational wave spacetimes.

Yes. The OP in post #12 appeared to be including those by mentioning "planar wave solutions".

 

[Pyter]

Other important vacuum solutions [...] include Schwarzschild spacetime and Kerr spacetime

I thought that those implied the presence of a mass. Weren't they found in relation with stars/black holes, even though they hold true in the empty space outside the fluid?

 

[PeterDonis]

I thought that those implied the presence of a mass.

No, they don't. They are vacuum solutions. Most physicists believe that these solutions will only describe a portion of spacetime, not all of spacetime, but that's a judgment about physical reasonableness, not a mathematical requirement of the solution. See further comments below.

A general property of solutions of the EFE that you might not be taking into account here is that any solution of the EFE describes an open region of spacetime. If all of spacetime is an open region, then a solution can describe all of spacetime. But any solution can also describe an open region of a spacetime that also contains other regions described by different solutions, where the solutions are matched at the boundaries of the regions according to certain constraints (called "junction conditions" in the literature).

So no solution by itself can ever "imply" the presence of particular other solutions. There are always multiple ways to construct global spacetimes from one or more EFE solutions, either by having a spacetime that is an open region which can be covered by a single solution, or by matching together multiple open regions in which there are different solutions.

Weren't they found in relation with stars/black holes, even though they hold true in the empty space outside the fluid?

No. Those solutions were originally discovered by simply doing math to solve the EFE under certain assumptions.

Schwarzschild, who discovered the vacuum solution that now bears his name, also discovered a solution for the interior of a spherically symmetric massive body, and it was not difficult to show that those two solutions can be fit together at the boundary of the massive body, so the vacuum solution describes spacetime outside the body. However, he did that only after discovering the two solutions separately.

We actually do not have any known construction of the interior of a rotating body that matches at the boundary to Kerr spacetime. The only thing we know for sure that Kerr spacetime describes is a rotating black hole.

 

[vanhees71]

Though it's not usually explicitly stated in textbooks, it's good to remember, what to do to define observable quantities within a gauge theory. A gauge theory is a theory, where you use more "computational degrees of freedom" than in principle necessary, and that's why the description has some redundancy.

E.g., in classical electrodynamics you use the four-vector potential to describe the observable electromagnetic field (with electric and magnetic components). The equations of motion for these observable fields are Maxwell's equations, and you introduce the four-potential to fulfill the two homogeneous ones identically. So you have to bother only about the inhomogeneous Maxwell equations in terms of the four-potential, but these equations do not uniquely determine the four-potential, because the four-potential is defined only modulo a gauge transformation. So to solve the inhomogeneous equations you can fix the gauge (in this case by imposing one gauge-fixing constraint like choosing the Coulomb or Lorenz gauge). That shows that the four-potentials do not have a clear physical meaning as observables but you have to calculate the electromagnetic field, which is observable, and these do not depend on the choice of gauge to solve for the four-potential. That's clear, because the original Maxwell equations for the observable fields have a unique solution (given initial and appropriate boundary conditions to solve them).

Now also GR is a gauge theory. The gauge freedom is the general covariance, i.e., the invariance of all observable quantities under arbitrary diffeomorphisms between coordinates.

Now in physical term's what's observable are usually local observables, e.g., the cosmic background radiation being detected by the Planck satellite. You can define observables most easily by measuring local scalar, vector, or tensor quantities (like the temperature of the background radiation as a function of direction as the Planck satellite most successfully did). To that end you introduce a local inertial reference frame in terms of choosing an appropriate tetrade (in this case fixed at the free-falling satellite). Then you can define your observables in a way that is independent of the choice of coordinates.

BTW: Historically this was the greatest obstacle for Einstein to find the final and very successful form of his GR. He had already the right idea concerning general covariance and also the field equations but then thought that this is not the right thing yet, because the solutions are not unique. It took him some more years to get back to these right equations and understand the physical meaning of the theory. This has been figured out in great detail in

J. Renn (Ed.), The Genesis of General Relativity, 4 vols., Springer (2007)

 

[Pyter]

To that end you introduce a local inertial reference frame in terms of choosing an appropriate tetrade (in this case fixed at the free-falling satellite). Then you can define your observables in a way that is independent of the choice of coordinates

How so? The position and time of the observable mass/energies still depend on the IFR's choice.

 

[ergospherical]

Those solutions were originally discovered by simply doing math to solve the EFE under certain assumptions.

I'd hesitate to describe the derivation of the Kerr metric as simple.

 

[vanhees71]

How so? The position and time of the observable mass/energies still depend on the IFR's choice.

Of course the components of four-vectors (the energy is a temporal component of the four-momentum of the particle) depend on the basis. A vector does not depend on the basis (momentum is a four-vector and thus basis independent). Mass is a scalar and thus basis-independent anyway.

 

[PeterDonis]

I'd hesitate to describe the derivation of the Kerr metric as simple.

Well, it turned out to be pretty simple once Roy Kerr did it.

 

[Pyter]

Just when I thought I had got it, this topic eluded me again.
I stated earlier that initially you take a snapshot of the stress-energy tensor in curvilinear coordinates to solve the EFE.
But it occurred to me that you can't do that because you don't know the curvature yet, not until you solve the EFE for $G^{ \mu \nu }$.
As written by @vanhees71, you initially assign the spacetime coordinates of the masses/flows in your extended region assuming you're in an in IFR (flat space).
But then your representation of the SET is inaccurate, because you can't account for aberrations in the positions (and times) due to the space curvature, like gravitational lens effects.
So your first solution of the EFE can't be the correct one.
And once you get a first solution for $G^{\mu \nu}$, do you repeat the process with your new SET coordinates until the solution converges?
Is this what the ADM formalism somehow does?

(As an aside, once you have a solution for $G^{\mu \nu}$, can you univocally determine the mapping of your curved spacetime to Euclidean spacetime, in order to be able to use curvilinear coordinates?)

 

[vanhees71]

It depends on what you want to do. The point is that it's not so simple to find solutions of these highly nonlinear equations of motion. If you have only continuum mechanical sources (usual simple models are "dust matter" or "perfect fluids"; already viscous fluids are not so simple anymore) if you have found a solution of the EFEs you also have a solution for the fluid motion, because the Bianchi identity ensures that $\nabla_{\mu} T^{\mu \nu}=0$, and these are the equations of motion for the matter. If you have in addition electromagnetic fields you have to solve for the Maxwell equations simultaneously.

If you look in textbooks with exact, analytic solutions like the Schwarzschild, Nordstrom-Reissner, Kerr, Friedmann-Lemaitre-Robertson-Walker metrics, the idea is just to make an ansatz for the metric components under assumptions of symmetries, choosing corresponding coordinates and solve for the EFEs with a correspondingly parametrized energy-momentum tensor (assuming also an equation of state or the fluid).

For numerical solutions you need to set up an initial-boundary-value problem with some foliation of spacetime. For that see

L. Rezzolla and O. Zanotti, Relativistic hydrodynamics,
Oxford University Press, Oxford (2013),
https://dx.doi.org/10.1093/acprof:oso/9780198528906.001.0001

 

[Ibix]

But it occurred to me that you can't do that because you don't know the curvature yet,

I believe that the usual approach for arbitrary spacetimes (e.g. when trying to predict gravitational wave signatures for LIGO) is to write down an approximation to the metric (e.g., model a pair of black holes as some sort of linear superposition of Kerr metrics) then use numerical methods to refine that to an actual solution on some spacelike slice, then evolve it. Large computers needed.

 

[Pyter]

It depends on what you want to do

Let's suppose I want to solve the EFE for the whole solar system, from the Earth's (or a geostationary satellite's) vantage point. There is no fluid, only (approximately) point-like masses whose 4D coordinates I observe through light beams or radio waves.

 

[vanhees71]

That would be "dust" ;-)).

 

[Pyter]

@vanhees71 ok.
Another question came to my mind from what you just wrote about the motion of the SET constituents.
Given that to get a "snapshot" of the SET in your desired spacetime region you have basically to track the planets' positions during the time interval delimiting your region, say from $t_{start}$ to $t_{finish}$, the net effect is that you already have determined, through observation, the equations of motions of the "dust".
So what would be the purpose of solving the EFE? Just to verify that, with the found $G^{\mu \nu}$, the EOM are geodesics (assuming only gravitational forces act on the masses)?

 

[vanhees71]

For dust you automatically get that the trajectories of the corresponding particles are geodesics. That's because "dust" by definition is just "free streaming", i.e., non-interacting particles:

See the solution of the 2nd problem on the following problem sheet:

https://itp.uni-frankfurt.de/~hees/art-ws16/lsg05.pdf

 

[PeroK]

Let's suppose I want to solve the EFE for the whole solar system, from the Earth's (or a geostationary satellite's) vantage point. There is no fluid, only (approximately) point-like masses whose 4D coordinates I observe through light beams or radio waves.

The first approximation of the solar system is to have the Sun as the only gravitational mass and the planets as "test particles". That gets you started and provides a test of the theory in terms of the correction to the precession of Mercury. However, most of the solar system dynamics are approximately unchanged from the Newtonian approximation. We can obtain approximate Schwarzschild coordinates from the distances obtained by previous observation.

To go further must become a very complicated task. I found this paper, for example:

https://arxiv.org/abs/1607.06298

There must be techniques of finding ever closer numerical approximations. I don't know if that addresses your question.

Note that if the solar system had some bizarre geometry that could hardly be guessed at, then the task of reconciling observational data with a theorectical model would be much harder. But, we know it's approximately Schwarzschild geometry and that gives us something to start from.

 

[Pyter]

For dust you automatically get that the trajectories of the corresponding particles are geodesics. That's because "dust" by definition is just "free streaming", i.e., non-interacting particles:

That doesn't model the general case, because the planets are also sources of gravity.

The first approximation of the solar system is to have the Sun as the only gravitational mass and the planets as "test particles"

You mean that you only include the Sun in the SET, compute the curvature, then find the equations of motions of planets imposing they're geodesics? Yes it surely is an approximation, but how would you refine it further to include the planets as sources of curvature? The paper you linked seems to contain a methodology for that.

The idea I've got so far is that a "trivial" (or even "feasible") solution of the EFE is only possible with a SET containing a single source of gravity. In that case, you place it at the origin of the coordinate system and compute the curvature of the space around it as if the other lesser masses were zero.

That also means that it's hard to solve the EFE in any spacetime region where only comparable masses are present.

 

[PeterDonis]

I stated earlier that initially you take a snapshot of the stress-energy tensor in curvilinear coordinates to solve the EFE.
But it occurred to me that you can't do that because you don't know the curvature yet, not until you solve the EFE for $G^{ \mu \nu }$.

You don't need to know the curvature to set up coordinates. Coordinates are just arbitrary numbers labeling points in spacetime. There is no requirement that they have to somehow "match up" with anything else.

In spacetimes with particular symmetries, it is usually helpful to choose coordinates that match those symmetries, but that still doesn't require you to know the curvature, just the symmetries.

(As an aside, once you have a solution for $G^{\mu \nu}$, can you univocally determine the mapping of your curved spacetime to Euclidean spacetime, in order to be able to use curvilinear coordinates?)

Coordinates are not a "mapping of curved spacetime to Euclidean spacetime".

 

[PeroK]

The idea I've got so far is that a "trivial" (or even "feasible") solution of the EFE is only possible with a SET containing a single source of gravity. In that case, you place it at the origin of the coordinate system and compute the curvature of the space around it as if the other lesser masses were zero.

That's patently not the case. Binary star systems are studied, for example.

 

[Pyter]

You don't need to know the curvature to set up coordinates. Coordinates are just arbitrary numbers labeling points in spacetime.

Coordinates are not a "mapping of curved spacetime to Euclidean spacetime"

Bear with me @PeterDonis, I'm sure you'll correct me if I'm wrong.
What I know about curvilinear coordinates system, i.e. to make a concrete example locating points on the surface of a 3D sphere (that is a 2D curved surface immersed in a 3D space), is that you map them to Cartesian coordinates. For the spherical surface you could use latitude and longitude, or the projection from one of the poles to a Cartesian 2D plane, and so on. In both cases you have an invertible map that biunivocally connects a point on the curved N-dimensional manifold to a point in the N-dimensional Cartesian space.
In that sense I've stated that to use curvilinear coordinates, you need a "map" to/from an Euclidean spacetime (maybe a "Cartesian space" would have been clearer). You use a point in the Cartesian space to identify a point on the manifold, through the (inverted) map.
What you call "arbitrary numbers" depend nevertheless from the choice of the coordinate system, i.e. where I place the origin, how I measure the distance, if I use straight "rulers" (radio/light waves) or not.

That's patently not the case. Binary star systems are studied, for example.

Perhaps, but for three or more bodies I bet things get more difficult. OTOH, the three-body problem is notoriously hard to solve even in Newtonian physics.

 

[Nugatory]

Let's suppose I want to solve the EFE for the whole solar system, from the Earth's (or a geostationary satellite's) vantage point. There is no fluid, only (approximately) point-like masses whose 4D coordinates I observe through light beams or radio waves.

If by “solve the EFE for the whole solar system” you mean “write down the metric in closed form, including the gravitational effects of the planets” you can’t. We can’t even find an exact closed-form solution to Newton's equation for the three-body problem and you're talking about a ten-body EFE problem. But the next sentence suggests that you don't really want to solve an intractably hairy problem of multiple coupled nonlinear differential equations - you want to get to a sufficiently accurate solution using coordinates in which the Earth is at rest.

The general recipe is the same whether you use GR or classical gravitation: First, find coordinates in which the planetary motion can be easily solved to the level of accuracy you need. A heliocentric frame in which the sun is at rest and the planetary masses are negligible may be good enough (and is the practical choice using GR - that's how we deal with the nomalous precession of Mercury). Second, find a coordinate transformation from your original coordinates to the coordinates you want, and apply it. Note that for the solar system this is already a solved problem - you can look online to see which planets will appear where in the night sky at any given time.