Exploring the Newtonian Limit of LCDM in Astrophysics: A Comprehensive Overview

[smallphi]

Is there an article about the Newtonian limit of LCDM?

The LCDM vacuum is deSitter not Minkowski and they have to expand around that so I think nonzero cosmological constant will make a difference. Give references if you know anything.

 

[pervect]

I haven't seen any such papers, but I'll make a few comments.

You can probably use MTW's approach to the Newtonian limit (pg 412) if you restrict the particle to be in some ring around the centeral mass M. Your test particle never gets "too close", so its escape velocity is always << c, and it never gets "too far away", either, so that the cosmological constant becomes unimportant.

Given that you restrict your particle to be in such a ring (not too close, not too far), and meet the other conditions that MTW discusses, you'll find that d^2 r / dt^2 is approximately (some constant) / r^2 as desired.

Note that you want g_uv = n_uv + h_uv, where h_uv << 1. The metric in http://arxiv.org/abs/gr-qc/0602002v2 will do that, but note that it is Schwarzschild-deSitter, not LCDM. (LCDM approaches a SdS if you wait long enough, but the later is stationary and the former isn't).

You'll basically get the usual result that g_00 = 1-2 Phi, but you'll have to throw away the term involving the cosmological constant to get a good match to Newtonian gravity. However, you can justify throwing away this term if you are within the correct range of distances (the ring I mentioned).

So, modulo some sign confusion, you'll get

Phi = M/r + Lambda*r^2/6

Note that $\nabla^2$Phi is not zero, i.e. you have to consider "empty" space as contributing to the gravitational field, as you might expect given a non-zero cosmological constant.

 

[smallphi]

I thought about that but what I need is exactly NOT throwing away the term with cosmological constant - I want to see what the correction is. I've seen this done for a particle moving on geodesic using the Fermi normal coordinates, and your example shows it for Schwarzschild - deSitter but was wandering if someone did it for some arbitrary mass density, not a point particle.

 

[pervect]

Well, if you take g_00 = 1 + 2Phi, where Phi = -M/r - Lambda*r^2/6

for the Schwarzschild-deSitter (SdS) solution, then in the Newtonian limit you have for the radial force component

Fr=M/r^2-Lambda*r/3

Thus the gauss Law integral of F*dA in some radius R is proportional to Fr*r^2, i.e

M -Lambda*R^3/3 = M - (4*Pi*R^3/3)*(Lambda/4*Pi) = M - V*Lambda/4*Pi

where V is the volume enclosed in a sphere of radius R.

so we see that the correct potential in the Newtonian limit is given by considering the Newtonian field of a mass M at the center, plus the Newtonian field of an infinite cloud of negative mass with a constant negative density per unit volume of -Lambda/4*Pi

This basically includes the effects of dark energy, and ignores the effects of dark matter, because of our choice of the SdS metric. If we want the Newtonian field for the LCDM model, we have to consider dark matter.

This is probably discussed in some paper somewhere, but I don't know where, maybe some other poster will contribute. That's basically the mental picture I use for the Newtonian limit in the absence of dark matter.

[add]I don't think the above method will work for the LCDM cosmology, unless you can find a form of the line element for LCDM where g_uv = n_uv + h_uv and h_uv << 1. Offhand, I'm not aware of any such coordinate chart (I suspect it may not even exist). Unfortunately it's necessary to make the formalism from MTW work. But you can do it for SdS, though I don't know of any particular paper that actually works it out.

 

[publius]

Pardon me for jumping in here, but since you all were so nice the last time, this brings something up about this g_00 = 1 + 2Phi(/c^2 if we're not using c = G = 1). As I understand it, that holds in the "weak field" limit, and if I were comparing clock rates say in the solar system, I could use that expression from the Newtonian Phi.

But I'm just not sure about the "weak field" and that factor of 2. For example, take Rindler's g_00, which is 1 + gz/c^2, where 'g' is the constant proper acceleration of our observer accelerating linearly in the +z direction. That takes the form 1 + Phi/c^2. But mear the origin, we still have something that looks like g_uv = n_uv + small h_uv. Here, our horizon occurs at z = -c^2/g. But if we take the Newtonian potential and solve for when the free fall speed reached light speed, we'd get just half of that.

So obviously, Newton and the 1 + 2Phi limit fails big time there. My gut tells me the reason is, if we're considering Rindler a "field", is that sucker ain't weak anywhere, we're on a big constant slope which we we've just gauged to one at our location, and it increases without limit with +z.

So does this mean our potential needs to go asympotically to 1 at infinity to get the factor of 2. But in the static deSitter plus Schwarszchild, that doesn't happen either, you've got that "cosmological horizon". But, there is no location with the potential gets much higher than 1. So is that what it is. No stationary clock should be running that much faster than our reference clock for this to work? :)

Again, forgive me for jumping in, but this is just one of those things that I puzzle over sometimes.

-Richard

 

[pervect]

Pardon me for jumping in here

Welcome aboard!

As I recall, Rindler's g_00 is (1+gz)^2, (with c=1), but I'd better double check that.

Yes, MTW, pg 173, aside from using different variable names, has the metric as

ds^2 = -(1+gz)^2 dt^2 + dx^2 + dy^2 + dz^2

(The original uses $\xi^0$ etc rather than t,x,y,z)

I think that removes this specific objection?

I have played a bit fast and lose with my analysis, it'd probably be good to look at it more closely to ensure that having h_uv << 1 in a small region is sufficient. As you point out, for large enough r, h_uv is not small in the SdS metric. At the current time I don't think this invalidates the analysis, but it does make it considerably longer if one is careful, and it is probably something that anyone who is interested in the problem should look at more carefully than I did in my post.

 

[publius]

DOH! That's right, it's squared, and would nicely expand to 1 +2gz + (gz)^2, giving us the 1 + 2Phi part for small gz, indeed. I forgot all about the squaring, there. Sorry about that.

-Richard

 

[publius]

Oh, here's something related to this, a couple of papers by Wolfgang Rindler himself and others and gravitational lensing in SdS:

http://arxiv.org/abs/0709.2948

http://arxiv.org/abs/0710.4726

Rindler says there was a misconception about gravitational lensing against a Lamdba background. In the SdS + Schwarzschild metric, Lambda cancels in the null geodesic equations, which led to the conclusion that Lambda had no effect on gravitational lensing. Rindler says "au contraire", here.

The problem was a coordinate misconception, assuming the asymptotic observer's local ruler and clock went to the t, r, theta, phi coordinates here as they do in Schwarzschild alone. But they don't, Lambda governs what they do far away, and Rindler shows it does effect the observed bending angle and goes on to show results in line with current estimates of Lambda.

-Richard

 

[smallphi]

I've seen the claims that Lambda doesn't influence gravitational lensing in SdS many many times. The problem is that one has to compare effects in two different spacetimes so one has to quantify the meaning of 'same' and 'different'. The claims I think are based on the false identification of the radial coordinates, r, in Schwarzschild and in SdS. They are just coordinates denoted by the same letter in two different spacetimes so they have different physical interpretation. Many people fail to see that.

Back to the main topic, Rindler's GR textbook 'Relativity: Special General and Cosmological' 2ed, says on page 304 that the Newtonian limit of GR with nonzero Lambda is

$$\nabla^2 \Phi = 4 \pi G \rho - \Lambda$$.

I have yet to see a clean derivation of that - in fact I haven't seen a signle derivation at all. Rindler glosses over it as if it is the same as the Lambda = 0 case and I don't think it is. I tried, starting from the 'isotropic' form of the SdS metric and it turns out, in order to get the result above, I need to throw away terms in the Ricci tensor that have an effect of the order of the effect produced by the Lambda term in the final equation. So I have to throw away terms but retain a term producing effect of the same order I've previously ignored - total nonsense.

Also, SdS is not asymptotically flat. How to understand the popular claim 'there is no Newtonian limit of spacetimes that are not asymptotically flat'. How to reconcile that with the above limit? To what length scales is the above limit correct (if its correct at all)? Is it applicable to galactic length scales for example?


Has anyone seen a paper on Newtonian limit of Schwarzschild-only. I want to see how they drop terms in the Newtonian limit there.

 

[pervect]

The way I read it, you can almost always get a limit where the approximate equations of motion of a particle are given by

$$\frac{d^2 x^i}{dt^2} = - \partial \Phi / \partial t$$

over some region R. These coordinates will be called "Gallilean" or "Newtonian" coordinates. It seems that it's usually assumed that R extends out to infinity, one might want to add in some disclaimers if R is (as in our example) restricted to R_min < R < R_max.

Since you don't appear to have MTW, I've attached a short scan for how you approximate the equations of motion to arrive at this conclusion. (I think this scan is small enough to be covered by fair use as far as copyright goes).

Note that MTW uses a comma to indicate normal partial derivatives. A semicolon (not used here) would be a covariant partial derivative.

Now, what do you call it when you have Newtonian coordinates, but Phi is not of the Newtonian form -M/r^2? I'm not quite sure how to describe this situation, which is the one you're interested in.

MTW's proof demonstrates that $\Phi = -h_{00}/2$ with the possible addition of an arbitrary constant to the potential, which won't affect the equations of motion.

 

[smallphi]

I have MTW. This is only half of the proof. The other half is the connection between Ricci(0,0), the matter density rho and the Christoffel in the geodesic equation which later is identified with the gradient of the potential. I already know all this. When they derive it, they first drop time derivatives of the metric because they consider a static case and second they drop any product of two Christoffels in Ricci(0,0) because each Christoffel is of first order in the metric perturbation when the unperturbed metric is Minkowski (for which the unperturbed Christoffels are all zero).That doesn't work when the background metric is de Sitter because its unperturbed Christoffels are not zero like Minkowski, and consequently the perturbed Christofells are NOT first order in the metric perturbation. That leads to extra terms in Ricci(0,0). Some of those terms create acceleration of the same order as the one created by the Lambda matter density in the final equation and it is inconsistent to drop them but leave Lambda.

MTW is not a good source for anything connected to Lambda because at those times they didn't know about 'Universe acceleration' and set Lambda = 0. Aside from that, even their Newtonian limit is not good cause they create the impression they do it for 'slowly moving' matter density when actually the Newtonian limit is for STATIC source. That is obvious from the final Newtonian equation - it describes INSTANTANEOUS force - if the matter density changes a little at one side, the gravitational potential responds instantaneously at some distant point which we know can't happen. So the derivation is for STATIC source not 'slowly moving' as MTW claim. It's not an accident that Sean Carroll and other textbooks explicitly talk about Newtonian limit for static source which initially puzzled me since MTW don't mention the word static at all.

 

[publius]

Aside from that, even their Newtonian limit is not good cause they create the impression they do it for 'slowly moving' matter density when actually the Newtonian limit is for STATIC source. That is obvious from the final Newtonian equation - it describes INSTANTANEOUS force - if the matter density changes a little at one side, the gravitational potential responds instantaneously at some distant point which we know can't happen. So the derivation is for STATIC source not 'slowly moving' as MTW claim. It's not an accident that Sean Carroll and other textbooks explicitly talk about Newtonian limit for static source which initially puzzled me since MTW don't mention the word static at all.

See this paper by Steve Carlip:

http://arxiv.org/abs/gr-qc/9909087

This was a response to some of Tom Van Flandern's "stuff" the "speed of gravity".

Carlip explicitly shows that GR gravity "extrapolates" motion to the second order, taking acceleration as well as velocity of the source mass. EM extrapolates only to first order. That is the E field points to where the source would be if it continued to move at constant velocity during the light travel meantime. GR gravity, takes acceleration into account as well, and will only "miss" if the source acceleration changes during the light travel meantime. This "missing" can be seen as the cause of radiation. If the forces deviate from equal and opposite, energy and momentum are "lost", and the radiation field must make up the difference.

So, while it may be a mess to pull out, the Newtonian limit is indeed good for "slowly moving" sources. That second order "extrapolation" makes the Newtonian potential act like it is responding instantaneously to changes in the source position. When things are moving/"jerking" fast enough for gravity to "miss" significantly, then you have gravitational radiation.

THe EM "extrapolation" is easier to see mathematically than GR, but the first order and second order differences between the two are difference between dipole and quadrapole radiation order. It's pretty slick as Carlip shows.


-Richard

 

[pervect]

I have MTW. This is only half of the proof.

OK, I didn't know you had it, or I wouldn't have bothered. This is the half I've got.

That doesn't work when the background metric is de Sitter because its unperturbed Christoffels are not zero like Minkowski, and consequently the perturbed Christofells are NOT first order in the metric perturbation.

Using the coordinate chart I mentioned for the SdS spacetime

$$ds^2 = (-1+2M/r+\Lambda r^2/3)\,dt^2+\frac{dr^2}{1-2M/r-\Lambda r^2/3}+r^2\,(d\theta^2+\sin(\theta)^2 d\phi^2)$$

I don't understand your point about the "unperturbed Christoffels". For instance, both the Schw metric and the above metric has various non-zero Christoffel symbols like, for instance, $\Gamma_{trt}$ nonzero, but so what? It doesn't affect the validity of MTW's argument. Nor can I find any non-zero Christoffel symbols in the SdS line element above which are not also non-zero in the Schw metric.

 

[smallphi]

So, while it may be a mess to pull out, the Newtonian limit is indeed good for "slowly moving" sources. That second order "extrapolation" makes the Newtonian potential act like it is responding instantaneously to changes in the source position. When things are moving/"jerking" fast enough for gravity to "miss" significantly, then you have gravitational radiation.

Of course GR takes into account finite speed of signals. The Newtonian gravity, which is the Newtonian limit derived by MTW, does NOT. It misses the corrections for the 'retarded position of the source' in that article. The MTW treatment is improper in the sense it silently drops such corrections without a hint or warning so at the end they got the Newtonian equations applicable to a static source not 'slowly moving with respect to light speed'.

 

[pervect]

I'm not convinced that MTW's treatment is "improper" either. I'll agree that a static metric has a timelike killing vector, and so has a conserved energy, while a non-static metic does not have a timelike killing vector, so their is no exactly conserved energy. However, I don't see what this has to do with the approximations that MTW is making or think that it makes them "improper".

MTW's approximation is like ignoring the magnetic field in a quasi-stationary electrostatics problem. There is no exactly conserved scalar potential in such a quasi-electrostatic problem with "slowly-moving" charges, but that doesn't stop it from being a good approximation.

BTW, the deSitter space-time (clarify: and the Schwarzschild-deSitter space-time) are both actually static, because they have time-like Killing vectors. (Another way of seeing this for the SdS space-time- inspect the metric coefficients I gave - none of them depend on 't'.)

Interestingly enough, according to http://www.bourbaphy.fr/moschella.pdf, deSitter actually gave the static coordinate chart first.

It is interesting to note that the first coordinate system used by de Sitter himself was a static coordinate system with closed spatial sections.

I did a little digging, and found the static coordinate chart for deSitter spacetime

ds2 = −F dT^2 + F^−1 dR^2 + R^2dS2, F = 1 − $\Lambda$R^2/3

online at
http://arxiv.org/PS_cache/gr-qc/pdf/9304/9304007v1.pdf

As one might expect, this is just the SdS spacetime with M=0.

So you can certainly carry out an analysis by both means for the deSitter space-time and the SdS spacetime, though it won't generalize to LCDM which doesn't have a timelike Killing vector.

 

[smallphi]

The question I had in mind is about the Newtonian limit of GR with non-zero cosmological constant i. e. Newtonian limits of its vacuum solutions deSitter and SdS. I guess I gave a wrong title of this thread.

I was finding the Newtonian limit of Schwarzschild to warm up. I considered a spherical star at the center of coordinates, a blob of matter far away from the event horizon (so no funky GR effects are present) and a test particle that feels the gravity of the star and the blob. I gave the Schwarzshild metric (isotropic form, rectangular coordinates) a small perturbing function, h_00(x,y,z,t), either of only g_00 or along the whole diagonal of the metric (mimicking scalar perturbation in cosmology) . I used Mathematica to compute the Ricci and Einstein tensors for the perturbed metric. Indeed Ricci_00 shows the expected term

$$-\nabla^2 h_{00}/2$$

but also terms like

$$\frac{ x^i \partial_i h_{00}} {r^2} \frac{M}{r}$$

I was not sure of the justification for neglecting the second type of terms - just saying M/r is small at the location of the blob is not enough, it depends how big the prefactor is. Finally, I figured out that each spatial derivative of h_00 is of order h_00/d where d = size of the matter blob. Assuming the blob size is much smaller than the distance between the blob and the star, I was able to drop the extra terms as insignificant.

The same result can be achieved much easier by linearizing the given metric around Minkowski and then adding a metric perturbation representing the blob. That can be done for Schwarzschild far from the center and for SdS in some intermediate radius, far from the center so M/r is small but not too far so Lambda r^2 is small too compared to 1. I noticed though that perturbing Minkowski that way produces pathological Ricci and Einstein tensors for which I will start a new thread.

Another thing bothers me about the Newtonian limit of SdS which is the main topic of this thread. The original de Sitter metric doesn't have a center - it's a homogeneous spacetime. Putting a blob of matter in it breaks that symmetry. The Newtonian limit shows that the cosmological constant is equivalent to repulsive force. In order to do calculations in Newtonian limit, we must know where the center of that force is, it is not equivalent to just take any point in the matter blob cause obviously that leads to very different predictions. That is of course equivalent to giving boundary conditions to the Poisson equation for the effective gravitational potential. I intuitively feel that after the matter blob settles in a stationary state, any gravitational waves go to infinity, the center of the repulsive force must coincide with the center of mass of the blob. That sounds very plausable but I can't give more convincing argument for it. I was hoping the analysis of the Newtonian limit will reveal why the center of the perturbed deSitter coordinates must be exactly at the center of mass of the blob but I've lost that hope. It sounds similar like asking why the metric around the spherical Sun is spherically symmetric - after all Schwarzschild is not the only possible vacuum solution. So the matter source somehow determines the final state of the metric around it. While that sounds OK in asymptotically flat spacetimes like Schwarzschild, in SdS it sounds strange that the very weak source M controls the very strong dS metric till infinity.