How to fill the stress energy tensor for multi body systems

[BiGyElLoWhAt]

Say I wanted to set up EFE for the earth and moon. How do I actually go about filling the stress energy tensor? I'm referencing the wikipedia page.
So the time-time should be approximately E/c^2V, so for the earth moon system
$T_{00} = \frac{3}{4\pi r_E^3}\frac{1}{c^2}(M_Ec^2 + 2/5 M_Er_E^2\omega^2)$
from 0 to r_E + [same for the moon] but from [center of the moon as a function of time] to [radius of the moon]
I guess my question is how do I rigorously add in these limits? So if I wanted to include the earth, sun, and moon, these limits on the location of the energy density/c^2 will be more noticeably important.

Certainly I don't have to write it as a fourier or taylor series, right? Right?

Any help is appreciated.

 

[PeterDonis]

How do I actually go about filling the stress energy tensor?

I would start by looking at a textbook instead of Wikipedia. (Carroll's online lecture notes discuss this some, for example.) If you don't already have some experience writing down stress-energy tensors for simpler situations, you're going to have a very tough time tackling this one.

For a good test case, I would start by trying to write down the stress-energy tensor for the interior of a single spherically symmetric, static object: i.e., nothing is a function of time (note: this assumes we have chosen appropriate coordinates), and nothing is a function of angular variables (again, this assumes we have chosen appropriate coordinates), so the only thing anything can be a function of is the radial coordinate $r$. This allows you to arrive at a form of the metric which only has two independent functions of $r$ in it; then you can compute the Einstein tensor for this metric, multiply it by $8 \pi$, and there's your stress-energy tensor.

 

[PeterDonis]

So the time-time should be approximately E/c^2V,

This is not a good way to proceed. How do you know your guess satisfies the Einstein Field Equation? Remember that the EFE says the SET is $8 \pi$ times the Einstein tensor. So to check whether any SET satisfies the EFE, you have to know the Einstein tensor. Guessing an SET is no help in figuring that out.

 

[PAllen]

Adding to @PeterDonis point, It is true that, mathematically, you can universally go the other way: pick an arbitrary metric, compute the Einstein tensor from it, and then check to see if the implied SET is physically plausible (e.g. energy conditions), and what it describes. However, a random rank 2 tensor field as a candidate SET has probability zero of being a possible Einstein tensor, because of the differential identities and integrability conditions that hold for the Einstein tensor.

 

[BiGyElLoWhAt]

I would start by looking at a textbook instead of Wikipedia.

Are you familiar with this book? It's the only one that I have.

 

[PeterDonis]

Are you familiar with this book?

Unfortunately I am not. Someone else here might be.

 

[Dale]

For a good test case, I would start by trying to write down the stress-energy tensor for the interior of a single spherically symmetric, static object:

@BiGyElLoWhAt I heartily second this recommendation

 

[pervect]

Say I wanted to set up EFE for the earth and moon. How do I actually go about filling the stress energy tensor? I'm referencing the wikipedia page.
So the time-time should be approximately E/c^2V, so for the earth moon system
$T_{00} = \frac{3}{4\pi r_E^3}\frac{1}{c^2}(M_Ec^2 + 2/5 M_Er_E^2\omega^2)$
from 0 to r_E + [same for the moon] but from [center of the moon as a function of time] to [radius of the moon]
I guess my question is how do I rigorously add in these limits? So if I wanted to include the earth, sun, and moon, these limits on the location of the energy density/c^2 will be more noticeably important.

Certainly I don't have to write it as a fourier or taylor series, right? Right?

Any help is appreciated.

What you'd actually do most likely is to use a post-newtonian approximation (PPN) https://en.wikipedia.org/wiki/Post-Newtonian_expansion, rather than the full theory. I don't really recall the details, but it's discussed in a lot of texts, such as MTW's "Gravitation". And the wiki article of course, though I don't use PPN enough to know how accurate the Wiki article on it is.

There's been a lot of refinements since the discussion in those older texts anyways - not in the theory itself so much, but making the theory usable in practice with Earth-based measurements of Earth-based atomic time, right ascension, and declination. There's a long and very technical exposition in the IAU 2000 resolutions and their various and fairly numerous ammendemnts, in which the IAU define a pair of coordinate systems (the BCRS and the GCRS) by specifying a metric for each , and additoinally a way to convert (approximately but with a high degree of accuracy) from one set of coordinates to the other.

As far as the stress-energy tensor goes, the stress part of the tensor doesn't directly contribute appreciably to the gravitational field of the Earth, but a point mass and/or spherical model for the distribution of matter in the Earth is a poor approximation, one needs to do a multipole expansion. Eliminating the pressure terms means that only the density (and momentum) terms are really important, but the effects of the Earth not being spherical make this not as simple as it seems.

 

[BiGyElLoWhAt]

... but a point mass and/or spherical model for the distribution of matter in the Earth is a poor approximation, one needs to do a multipole expansion. Eliminating the pressure terms means that only the density (and momentum) terms are really important, but the effects of the Earth not being spherical make this not as simple as it seems.

Do you know what they used to get this? Is it empirically calculated?

An example is the calculation of the rotational kinetic energy of the Earth. As the Earth has a period of about 23.93 hours, it has an angular velocity of $7.29×10^{−5} rad/s$. The Earth has a moment of inertia, $I = 8.04×10^{37} kg·m^2$.[1] Therefore, it has a rotational kinetic energy of $2.138×10^{29} J$.

 

[PAllen]

Do you know what they used to get this? Is it empirically calculated?

That looks like a pure Newtonian calculation.

 

[pervect]

Do you know what they used to get this? Is it empirically calculated?

No, I don't. I know that JPL Ephermedies does take into account the figure of the Earth, and also has a model to include the effects of Earth and lunar tides to calculate the orbital effects for the ephermis, but I don't know for a fact if the JPL Ephermedies (of which there are a bunch of versions) uses a simple ellipsoid model for the Earth's figure, or something more sophisticated. I seem to recall skimming a theory paper by the JPL group at one time, but I couldn't find it again.

From what I read, lunar laser rangefinding has been used to get a better model of the tide-induced part of the pertubation, but it's felt that it's only safely applied for times relatively close to our own era.

 

[pervect]

I did find a link on the JPL ephermedies. https://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf covers de430 and de431. Unfortunately I'm not sure how well it answers your question in detail. A summary of my read on this is that the internal structure of the Earth, moon, planets, and sun does have an effect on their gravitational fields and orbital motions. (For instance, the Earth has an iron core, and the moon is believed to have one as well).

What has been directly measured, and what has been fit to make the simulations match the observations isn't really clear to me.

It appears that the Earth's moon has the most complex structure as far as uneven distribution of mass goes. I believe I've heard this referred to as "lunar mass concentratoins", sometimes abbreviated, and it was historically important to the Apollo missions. This is important because of the tight coupling between the Earth and the moon, and because our observations are (mostly?) Earth-based, so we need to know accurately how the Earth moves and how it's axis of rotation changes (precession of the equinoxes).

Some of the details:

III. Translational Equations of Motion

The translational equations of motion include contributions from: (a) the point mass interactions among the Sun, Moon, planets, and asteroids; (b) the effects of the figure of the Sun on the Moon and planets; (c) the effects of the figures of the Earth and Moon on each other and on the Sun and planets from Mercury through Jupiter; (d) the effects upon the Moon’s motion caused by tides raised upon the Earth by the Moon and Sun; and (e) the effects on the Moon’s orbit of tides raised on the Moon by the Earth.

....

A. Point Mass Mutual Interaction

The gravitational acceleration of each body due to external point masses is derived from the
isotropic, parametrized post-Newtonian (PPN) n-body metric [24–26].

....

B. Point Mass Interaction with Extended Bodies

The modeled accelerations of bodies due to interactions of point masses with the gravitational field of nonspherical bodies include: (a) the interaction of the zonal harmonics of the Earth (through fourth degree) and the point mass Moon, Sun, Mercury, Venus, Mars, and
Jupiter; (b) the interaction between the zonal, sectoral, and tesseral harmonics of the Moon
(through sixth degree) and the point mass Earth, Sun, Mercury, Venus, Mars, and Jupiter;
(c) the second-degree zonal harmonic of the Sun (J2) interacting with all other bodie

An image I found that might help explain this (I had to look up the terminology).

So the mathematical tool used to handle the distribution of mass in the planets (and moon) is spherical harmonics.

I think I first ran into the spherical harmonics in the context of gravity in Goldstein's "Classical Mechanics" in the section on potential theory. Goldstein used the Earth-moon system as an example of potential theory. This was all in the context of Newtonian mechanics though.

So the PPN theory is a theory of point masses, and on top of this additional , basically Newtonian, corrections due to spherical harmonics of the gravitatioanl fields due to the uneven distribution of matter (including, but not limited to, the equatorial bulges of spinning objects) is added in as needed.

I think DIxon has a more formal treatment for extended bodies in GR, but while I know it exists, I'm not really familiar with the details. There may be better papers on the topic of extended bodies in GR than Dixon's as well. http://rspa.royalsocietypublishing.org/content/314/1519/499

 

[BiGyElLoWhAt]

@PeterDonis What do you mean by SET? Is there a set of constraints that I'm missing? I've tried looking around, I watched some videos on the stress-energy tensor as well as some google searches, but didn't find anything other than "constraint field theories" which I don't think is applicable. I could be wrong about this, though. My "gut" tells me that I should be able to either set up a fourier series, taylor series, or set of functions of spheres, derived relativistically, and have that work. If not, could you briefly explain what you mean by satisfying the SET, as you put it?

 

[PeterDonis]

What do you mean by SET?

Stress-energy tensor.

I watched some videos on the stress-energy tensor as well as some google searches

You marked this thread as "A" level. That means your knowledge of what the stress-energy tensor is should not be based on videos and google searches. It should be based on textbooks and peer-reviewed papers. If you don't have that background, then the problem you posed for yourself in the OP is beyond your current level of knowledge.

 

[PeterDonis]

@BiGyElLoWhAt Based on post #13, I have changed the level of this thread to "I". However, even that level requires more background than videos and google searches. But the problem you posed in the OP is not discussible at the "B" level; it's too advanced. Even the "I" level is possibly marginal.

 

[BiGyElLoWhAt]

I was more looking for what you meant by SET with the searches. For some reason I didn't see the acronym in it.

 

[BiGyElLoWhAt]

But yes, my knowledge of the SET stems mostly from susskinds GR lectures, the book I linked, and other similar resources.

 

[PeterDonis]

my knowledge of the SET stems mostly from susskinds GR lectures, the book I linked, and other similar resources.

Does the book you linked to discuss how to solve the Einstein Field Equation for a simple case like a spherically symmetric massive object?

 

[BiGyElLoWhAt]

There is the stress energy tensor of a particle with a trajectory, which I would assume is the foundation for all other SET's.
*
It immediately goes into wave solutions of EFE.

 

[BiGyElLoWhAt]

... Therefore, the field equations (4.1) necessarily require that
$T^{\mu\nu}_{;\nu} = 0$ (4.14)
What is the significance of the above relation? The corresponding relation in special relativity, namely $\partial \tilde{T}^{\mu\nu}/\partial x^{\nu} = 0$ , represents, as we all know, the conservation of energy and momentum. But in its present form (4.14) does not refer to any conservation law at all. In fact, in the presence of gravitation, the energy and momentum of matter alone is not even conserved. What (4.14) does represent is the equation of motion. To see this result, let us consider the energy-momentum tensor of a single particle of rest mass equal to m
$T^{\mu\nu}(x) = \frac{m}{\sqrt{-g}} \int U^{\mu}U^{nu}\delta^4(x-y(s))ds; U^{\nu} = \frac{dy^{\nu}}{ds}$ (4.15)

It uses covariant divergence to obtain a homogeneous equation and and 4.14
$\int U^{\mu}U^i\delta(x^0-y^0)\frac{\partial}{\partial x^i}\delta^3(x-y)ds = -\int U^{\mu}\delta(x^0-y^0)\frac{\partial}{\partial y^i}\delta^3(x-y)dy^i = -\int U^{\mu}\delta(x^0-y^0)\frac{\partial y^i}{\partial y^0}\frac{\partial}{\partial y^i}\delta^3(x-y)dy^0$ ... (more simplification)
and come to the result $\int (\frac{dU^{\mu}}{ds} + \Gamma^{\mu}_{\lambda \nu}U^{\lambda}U^{\nu})\delta^4(x-y(s))ds = 0$

"from which the equation of motion (3.16) follows by inspection."

I think I typed everything correctly. I found 1 typo while transcribing this, so it's possible there are more.

... The fact that the EFE predict the equation of motion is quite remarkable and may be contrasted to the situation in electrodynamics, where the Maxwell's equations do not contain the corresponding equation of motion. The origin of this distinction lies in the non linear character of the Einstein equations. The physical significance of this nonlinearity resides in the fact that gravitational fields carry energy while, for instance, electromagnetic fields do not carry charge.

4.3 gravitational waves
...

 

[BiGyElLoWhAt]

I guess that doesn't really derive the tensor, though. It kind of just gives it.

This is the first chapter that T shows up in, and it's introduction is given in EFE. It immediately goes into studying consequences of the EFE. 4.1 is The newtonian limit, 4.2 (see previous post) is Equation of motion,4.3 is Gravitational waves, 4.4 generally symmetric gravitational fields, then chapter 5, the swartzchild line element, (it's given, not derived).

 

[PeterDonis]

There is the stress energy tensor of a particle with a trajectory, which I would assume is the foundation for all other SET's.

No, it isn't. You still haven't answered my question: does your textbook discuss the case of a spherically symmetric massive body? That's a good starting case to work out before you try anything more complicated.

It immediately goes into wave solutions of EFE.

Where are you getting that from?

 

[BiGyElLoWhAt]

No it doesn't have any worked examples like that. The next section in the chapter is on g waves in the weak field limit and matter free regions.

 

[PeterDonis]

it doesn't have any worked examples like that

Then it's not going to help you with the problem you posed for yourself in the OP, unless you want to treat the Earth and Moon as point masses, in which case the post-Newtonian frameowork that @pervect has already referred to is what you will end up using. In this framework the stress-energy tensor isn't even used; you're basically treating each mass as a point mass, whose only meaningful parameters are its mass and multipole moments, and computing the vacuum Einstein Field Equation in the regions between the masses (more precisely, you're approximating the vacuum EFE in the weak field limit).

 

[BiGyElLoWhAt]

My goal is to calculate the equations of motion of the solar system and compare them to empirical data, so the point mass approximation, I think, is no good. Can you recommend a text to help with this? Books are fine.