Is There a Universal Redshift Formula for Arbitrary Spacetime Metrics?

[tom.stoer]

Quick question: is there a general formula for the redshift z_C[g] for a photon traveled in a spacetime with arbitrary metric g along a light-like geodesic C?

I do neither want to use a special symmetry for g (spherical, homogeneous and isotropy) nor do I want to use a specific expression for g, like FRW metric or whatever.

 

[marcus]

That's a nice question! The Bunn Hogg paper would not exactly address it because I think they were specializing to the FRW case. But they might have some expression that could be usefully adapted.

http://arxiv.org/abs/0808.1081
The kinematic origin of the cosmological redshift
Emory F. Bunn, David W. Hogg
(Submitted on 7 Aug 2008 (v1), last revised 14 Apr 2009 (this version, v2))
A common belief about big-bang cosmology is that the cosmological redshift cannot be properly viewed as a Doppler shift (that is, as evidence for a recession velocity), but must be viewed in terms of the stretching of space. We argue that, contrary to this view, the most natural interpretation of the redshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts. The stretching-of-space interpretation obscures a central idea of relativity, namely that it is always valid to choose a coordinate system that is locally Minkowskian. We show that an observed frequency shift in any spacetime can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon's path. In the context of the expanding universe the kinematic interpretation corresponds to a family of comoving observers and hence is more natural.
6 pages. Am.J.Phys.77:688-694,2009

One equivalent way to analyze the redshift is as the cumulative effect of a large (infinite) number of Doppler shifts along the path.

 

[tom.stoer]

Hi marcus, thanks, but the paper doesn't mention anything like that.

The most general formula I know is for the Robertson-Walker line element

d\tau^2 = dt^2 - a^2(t) \, \left( \frac{dr^2}{1-kr^2} + r^2\,d\Omega^2 \right)

where spherical symmetry and isotropy has been used, but where the scale factor a(t) is still an arbitrary function of time.

It says that for light emitted from a co-moving source and received by a co-moving observer the redshift z is given by scale factors

1+z = a_0 / a_1

 

[tom.stoer]

What I am looking for is something like

z_C[g] = \int_C ds \, f[g]
which can be applied to arbitrary metrics g and arbitrary (light-like) geodesics C.

EDIT: wait, this can't be fully correct, b/c this eq. looks covariant, whereas frequencies and redshifts are source- and observer-dependent; that means it must be something like

z_{C,S,O}[g]
but now I am even more confused about the r.h.s.

 

[fzero]

It's worthwhile to look back over the derivation of the redshift for FRW and see why you could get such an explicit formula in the first place. First, the null geodesic equation $ds^2=0$ is separable. As a result, we can show that the scaled time-elapsed between emission and detection of successive crests of a wave is independent of positions. Then you use the approximation that the scale factor is constant over the period of the wave.

The FRW derivation does not require a solution of the geodesic equation, but it's clear that in the general case, you will need one. Obviously you're not going to get too far by keeping the metric arbitrary. You might get a bit further if you restrict to a class of metrics for which the null geodesic equation is separable, but these probably all be FRW-like, where we don't restrict to constant spatial curvature.

 

[Chalnoth]

I'm not sure it's possible to write down something quite that simple, unfortunately. There are a few steps that you need to take to do this with a general metric:
1. Find the path the light ray follows. In the simplest case, this can be simply by solving ds^2 = 0 (because light must follow null geodesics). If the metric is sufficiently complicated, this won't be enough, and you'll have to solve the geodesic equation, which is far more complicated. See here:
http://en.wikipedia.org/wiki/Geodesics_in_general_relativity

2. Once you've found the path the light ray follows, you can pick a suitable set of observers along that path in the way that Bunn & Hogg did in the paper Marcus linked above.

 

[tom.stoer]

Chalnoth, thanks for the comment, but what you are explaining is much more than I had in mind.

I do not want to find the geodesic C, I only want to have an expression which works for arbitrary, unspecified, light-like geodesics.

Simple example: Suppose I am asking for the general expression for the length of a geodesic C. The answer is simply

S = \int_C ds

where C remains unspecified! That's all I am looking for.

 

[fzero]

If you look at Wald (5.3.6) and the preceeding discussion, you'll find an expression for the redshift in the case that there exists a Killing vector. This is quite an improvement on the heuristic way the redshift is usually computed for FRW.

 

[tom.stoer]

Thanks, great, this is still not fully generic b/c it relies on the Killing vector, but it's an improvement compared to the usual derivation

 

[tom.stoer]

I thought about it, and I can't believe my results. Applying the idea of Bunn and Hogg the infinitesimal Doppler shift at two spacetime points separates by dr, dt is always

df/f = - H dt

But H is always

H = d ln(a) / dt

and therefore the diffential equation can always be integrated w/o ever solving for a geodesic. That means that given two spacetime points connected by a light-like geodesic and with co-moving time t the formula

1 + z = a(t₀) / a(t_e)

is always valid, regardless what a(t) is.

But this can't be correct.

The problem is that the definition of H(t) and a(t) cold make sense locally i.e. for infinitesimal Doppler shifts, but not globally b/c there we expect dependence on spatial coordinates.

Therefore the first equation has to be modified

df / dt = -H(r) dt

were we assume that for each infinitesimal Doppler shift the approximation of an RW line element is still valid locally. But I am afraid that this starting point with H(r) defined locally does not really make sense.

Any ideas?

 

[Chalnoth]

1 + z = a(t₀) / a(t_e)

is always valid, regardless what a(t) is.

But this can't be correct.

Yes, it is correct. The wavelength of photons is increased by the same factor as the expansion.

The problem is that the definition of H(t) and a(t) cold make sense locally i.e. for infinitesimal Doppler shifts, but not globally b/c there we expect dependence on spatial coordinates.

In order to have dependence upon spatial coordinates, you have to have a metric which changes in space. The FRW metric does not.

 

[Haelfix]

Tom is asking about a Doppler shift formula for a general metric.

It seems to me the problem is really one of choice. Namely, how do you want to compare two distant tangent spaces. There are many different geodesics that could be chosen, and hence there is no canonical choice. But it seems to me that you would want one that reproduces standard intuition. Namely you want a choice of geodesic that yields something that looks like the standard decomposition into cosmological redshift terms + Doppler shift terms.

I believe I worked this out a long time ago in a problem set or somesuch, but I think the general gist is at the very minimum, you need a globally hyperbolic spacetime that can be foliated by hypersurfaces of constant time. This fixes one of the degrees of freedom, and you can then parralel transport the four vectors along the induced geodesic (eg the mutual points of intersection along the worldlines of a chosen test particle).

I do not remember if there was additional arbitrariness (but most likely yes, eg you probably want some symmetry to kill off diagonal terms or you will have to make some choices about shear terms)

 

[tom.stoer]

If you look at Wald (5.3.6) and the preceeding discussion, you'll find an expression for the redshift in the case that there exists a Killing vector. This is quite an improvement on the heuristic way the redshift is usually computed for FRW.

Agreed.

But what are intesting spacetimes with Killing vectors?

 

[martinbn]

But what are intesting spacetimes with Killing vectors?

What do you mean? Kerr, FRW, aren't these interesting?

 

[tom.stoer]

No, they aren't interesting ;-) what I am really interested in are metrics w/o symmetries, generalizations of the derivation using Killing vector fields etc.

 

[Chalnoth]

No, they aren't interesting ;-) what I am really interested in are metrics w/o symmetries, generalizations of the derivation using Killing vector fields etc.

Right. That is horribly ugly and incredibly difficult to do properly. For reference, nobody has solved the Einstein equations for a metric that does not obey spherical symmetry. Granted, finding null geodesics is a simpler problem than that, but it is in no way simple for a general metric.

 

[George Jones]

For reference, nobody has solved the Einstein equations for a metric that does not obey spherical symmetry.

There are loads of exact solutions to Einstein's equation that aren't spherically symmetric.

 

[fzero]

Agreed.

But what are intesting spacetimes with Killing vectors?

In principle, given a sufficiently nice 3-manifold $M_3$, we can construct an Einstein 4-manifold as the total space of a circle bundle over $M_3$. This includes various gravitational instanton solutions, so I would say that they are "interesting." I don't have any more insight into the problem w/o a Killing vector.

 

[George Jones]

I thought about it, and I can't believe my results. Applying the idea of Bunn and Hogg the infinitesimal Doppler shift at two spacetime points separates by dr, dt is always

df/f = - H dt

But H is always

H = d ln(a) / dt

and therefore the diffential equation can always be integrated w/o ever solving for a geodesic. That means that given two spacetime points connected by a light-like geodesic and with co-moving time t the formula

1 + z = a(t₀) / a(t_e)

is always valid, regardless what a(t) is.

But this can't be correct.

Sorry, I'm having trouble understanding this. In the context of a general metric and coordinate systems (as in your original post) what are: H; a; co-moving t?

The shift depends not just on the locations of the emission and reception events, but also on the worldlines (i.e., 4-velocities) of the observers at emission and reception 4-velocities, as you said in a previous post.

Later this week I will try and post a calculation for an FRLW example (possibly using non-zero peculiar velocities), but instead of using a usual method like

https://www.physicsforums.com/showthread.php?p=3978731#post3978731

I will try and use a general method that applies to all metrics and coordinates systems.

I haven't done this, so I don't know if I will be successful, but, when I get some time, I would love to have a go at this.

 

[tom.stoer]

Sorry, I'm having trouble understanding this. In the context of a general metric and coordinate systems (as in your original post) what are: H; a; co-moving t ...

agreed - I only wanted to express my cluelessness.

I think the best way i to try to generalize Wald's method, i.e. the parallel transport of the projection of a wave vector k on a velocity u.

Another idea could be to use the Greens function for a general metric and to apply a reduction to geometric optics.

 

[Chalnoth]

There are loads of exact solutions to Einstein's equation that aren't spherically symmetric.

Sorry, I guess you're right. But from what I can tell, this hasn't been done in the cosmological context yet. This is what I understand, anyway, from the discussion surrounding the attempts to determine whether or not the accelerated expansion could be put down to improperly handling the inhomogeneity of our universe. Some researchers even went so far as to make use of spherically-symmetric models that varied only in the radial direction to investigate this question.

 

[fzero]

Sorry, I guess you're right. But from what I can tell, this hasn't been done in the cosmological context yet. This is what I understand, anyway, from the discussion surrounding the attempts to determine whether or not the accelerated expansion could be put down to improperly handling the inhomogeneity of our universe. Some researchers even went so far as to make use of spherically-symmetric models that varied only in the radial direction to investigate this question.

I've never studied them in any detail, but there are the Bianchi models, which are homogeneous, but not necessarily isotropic. The FRW are examples of specific classes of Bianchi models, as are the Kasner metrics, which are the most commonly studied "cosmological" solutions w/o spherical symmetry.

 

[Chalnoth]

I've never studied them in any detail, but there are the Bianchi models, which are homogeneous, but not necessarily isotropic. The FRW are examples of specific classes of Bianchi models, as are the Kasner metrics, which are the most commonly studied "cosmological" solutions w/o spherical symmetry.

As near as I can tell, neither of these are interesting in the sense that they can't be used to study the real anisotropies we observe. That is, they can be used to describe a universe that expands at different rates in different directions, but can't be used to study the inhomogeneities due to galaxy clusters.

 

[tom.stoer]

The most interesting cosmological models w/o symmetry are swiss cheese models which consist of patches of FRW models; the intention is to model large scale inhomogenities like voids and (at least partially) to explain observed redshift not as an effect of accelerated expansion but as an effect light propagation in non-FRW spacetime.

It is not interesting in this context b/c for each FRW-patch the standard formulas for redshift do apply.

 

[Haelfix]

The shift depends not just on the locations of the emission and reception events, but also on the worldlines (i.e., 4-velocities) of the observers at emission and reception 4-velocities, as you said in a previous post.

Yes and further there is an implicit choice here on how to compare the 4 velocities in the cosmological context. For a general metric without any symmetries, there is absolutely no way to solve this without both choosing how to parralel transport a chosen vector as well as choosing which geodesic to use. There is no god given choice, other than the one that seems to provide the right intuition(namely something that must look like a generalization of the standard decomposition into a doppler part + cosmological part... For a general choice, there will not be anything like this decomposition).

So I again don't see how you can possibly solve this without resorting to the geodesic equation.

 

[tom.stoer]

Let's define the general geometric context.

We have a geodesic C connecting two spacetime points P and Q. Then have two observers O_B and O_Q located in P and Q with 4-velocities u_P and u_Q. In P we have a tangent vector t_P of the geodesic C.

Now we define a point P' as

x_{P^\prime}^\mu = x_P^\mu + d\tau_P\,u_P^\mu

on the worldline of O_P. Starting at P' we define a new geodesic C' using the (transported) tangent vector t' in P'. And we define a new point Q' using the same construction at Q (this is not straightforward b/c the 4-velocity u_Q is not arbitrary b/c the worldline of O_Q must intersect C').

We identify the two geodesics C,C' starting at P,P' and ending at Q,Q' with two light signals. The "frequency" is replaced by the two proper time intervals, therefore the redshift can be defined via

\nu_Q / \nu_P = d\tau_P / d\tau_Q

Do I miss anything?

 

[Chalnoth]

The most interesting cosmological models w/o symmetry are swiss cheese models which consist of patches of FRW models; the intention is to model large scale inhomogenities like voids and (at least partially) to explain observed redshift not as an effect of accelerated expansion but as an effect light propagation in non-FRW spacetime.

Right, but as I understand it these sorts of attempts have failed.

 

[tom.stoer]

fzero at al.

If you look at Wald (5.3.6) and the preceeding discussion, you'll find an expression for the redshift in the case that there exists a Killing vector. This is quite an improvement on the heuristic way the redshift is usually computed for FRW.

I think I don't get Wald's argument completely; why DOES (5.3.1) depend on the 4-velocity, whereas (5.3.6) does NOT?

 

[martinbn]

Because the light 4-vector is null and you can decompose it as a sum of two parts along the 4-vector of the isotropic observer and perpendicular to it i.e. spacelike. k=k_u+k_\perp so 0=k\cdot k=k\cdot k_u +k\cdot k_\perp. These may be not his notations. The term k\cdot k_u is the one that contains the 4-velocity of the observer and that relates to the frequency, but up to a sign it is k\cdot k_\perp, the one that has the spacelike Killing vector.

 

[tom.stoer]

so he defines a special observer?