Is There a Universal Redshift Formula for Arbitrary Spacetime Metrics?

[tom.stoer]

Thanks a lot.

My impression is that Wald's method cannot be generalized b/c for fixed wave vector k at the Source it is unclear how to transport it along the null-geodesic. Reason is that we only have a fixed vector k, but what we really need is a vector field k(x).

I have to think about the derivation of geometric optics based on Maxwell-equations on curved spacetimes. For the moment I think a rather general setup would be a wave equation (Klein-Gordon for simplicity) and a solution

$$\phi(x) = e^{ikx}\,f(x)$$

where k is a constant 4-vector (in a certain spatial region) and f(x) is a slowly varying function (compared to the wavelength defined by k and the curvature radius). Then it makes sense to define a 'frequency operator' for an observer field with 4-velocity u(x)

$$\Omega_u = -i\,u^\mu(x)\,\partial_\mu$$

and a 'frequency'

$$\omega = \phi^{-1}\,\Omega\,\phi = u^\mu\,k_\mu -i\,u^\mu\,\partial_\mu\,\ln f$$

 

[Mordred]

you may have already looked at this option see attached. Not sure if the methods in the article will help but it may give you some further direction

 

[tom.stoer]

Thanks for the interesting link.

I think the most interesting result in the present context are eq. (41) and (42). The paper demonstrates how to derive well-known results when symmetries are present, and how to relate them to Doppler-shift. However, there is no general rule for the parallel transport of tangent vectors along the null geodesic. Or do I miss anything?

 

[Mordred]

The paper itself is one I had in my archives. It will take me some time to recall the specifics. I presented it as it may provide an alternate means of correlating redshift in a non FLRW metric fashion.
However it is an older methodology. I was cleaning out my archives when I thought of this post.
The usefulness to your criteria
will depend on your goal

 

[tom.stoer]

The goal is to derive a general formula (as suggested in post #4) which applies to arbitrary spacetimes with metric g(x) and an observer field with 4-velocity u(x) for two spacetimes points P and Q connected by a null-geodesic C. The idea is to "add up" infinitesimal redshifts along C, and to find an appropriate function f, i.e. something like

$$z_{C,u_P,u_Q}[g] = \int_C d\lambda \, f[g,u]$$

 

[tom.stoer]

Wait, I found a formula - unfortunately w/o proof - but it seems to be exactly what I am looking for.

They consider a concept like that proposed in post #26.

We have a null-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. Then we have an infinitesimally neighbored null-geodesic C' connecting two points P' and Q' on the worldlines of the observers, defined via their 4-velocities. 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 defined via the 4-velocities on the observer worldlines connecting P with P' and Q with Q', respectively.

Therefore the redshift can be defined via the proper times

$$z = \frac{d\tau_Q - d\tau_P}{d\tau_P}$$
$$1+z = \frac{d\tau_Q}{d\tau_P}$$

I think this is straightforward.

Now they introduce the null-geodesic $x^\mu(\lambda)$ with affine parameter $\lambda$ and claim that

$$1+z = \frac{\langle\dot{x},u\rangle_Q}{\langle\dot{x},u\rangle_P}$$
$$\langle x,y \rangle = g_{\mu\nu}\,x^\mu\,y^\nu$$

This seems to be fully generic but also rather strange - at least to me - b/c the redshift does not depend on the spacetime along C. Only the geometry at the two points P and Q is required. This was true for all special constructions considered so far (using Killing vectors, ...) but seems to be true for general situations w/o any symmetry, too.

EDIT: here' the reference, eq. (37) in http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&page=articlesu4.html

 

[Mordred]

This approach is fascinating thanks for the link. There is a lot in the article I don't fully understand but that's what makes it fun lol. I'm such a sucker for punishment lol

 

[tom.stoer]

Another formulation which represents the redshift as an integral over spacetime-expansion.

Consider a metric

$$ds^2 = dt^2 - g_{ik}\,dx^i\,dx^k$$

with a null-geodesic $x^\mu(\lambda)$ and tangent

$$n^\mu = (n^0,n^i) = \frac{dx^\mu}{d\lambda}$$

Then define

$$h(n) = \frac{\partial_t g_{ik}\,\,n^i\,n^k}{g_{ik}\,n^i\,n^k}$$

One finds

$$\ln(1+z) = \int_{t_P}^{t_Q}dt\,h(n)$$

Refer to http://sdcc3.ucsd.edu/~ir118/Leiden2010/redshift-nat-final-journal.pdf

The overall idea of the paper may seem a little strange, but the above mentioned formula is interesting. However, I was not able to find this formula anywhere else, so we should better double-check its derivation.

 

[tom.stoer]

the longer I try to understand the proof, the more obscure the paper seems to be ...

 

[Bobbywhy]

Thanks for the interesting link.

I think the most interesting result in the present context are eq. (41) and (42). The paper demonstrates how to derive well-known results when symmetries are present, and how to relate them to Doppler-shift. However, there is no general rule for the parallel transport of tangent vectors along the null geodesic. Or do I miss anything?

Does either of these papers contain the information you search for?

General Relativity and Quantum Cosmology
Title:Averaged null energy condition in a classical curved background
Authors: Eleni-Alexandra Kontou, Ken D. Olum
Abstract: The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative.
arXiv.org > gr-qc > arXiv:9509004

General Relativity and Quantum Cosmology
Title:The Singularity Problem for Space-Times with Torsion
Authors: Giampiero Esposito
Abstract: The problem of a rigorous theory of singularities in space-times with torsion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport.
arXiv.org > gr-qc > arXiv:1212.2290

 

[WannabeNewton]

Tom, I'm sorry to bring up an old thread and I haven't exactly looked through all the posts in this thread but have you seen if you could generalize the method of problem 5.4 in Wald?

 

[tom.stoer]

Tom, I'm sorry to bring up an old thread and I haven't exactly looked through all the posts in this thread but have you seen if you could generalize the method of problem 5.4 in Wald?

Thanks for asking.

Yes, I think the generalization as described in post #41 is exactly what I was looking for. I have to find the old references (Brill, Schroedinger and Straumann, not available online) in order to understand the proof. It seems to be a fully generic formula for the redshift of a photon along its geodesic in an arbitrary spacetime. The redshift is fully encoded in the geodesic x^μ(λ) and the observer field u^μ.

That's a nice result.

 

[WannabeNewton]

The redshift is fully encoded in the geodesic x^μ(λ) and the observer field u^μ.

That's a nice result.

Ah I see, so I was just late to the game ! Thanks for the link as well as that is quite a result indeed!

 

[George Jones]

Tom, I'm sorry to bring up an old thread and I haven't exactly looked through all the posts in this thread but have you seen if you could generalize the method of problem 5.4 in Wald?

As I said, Wald treats the general case

The second paragraph of 5.3, however, is completely general. This paragraph outlines the general method I had in mind when I wrote my previous post.

Wald's equation (5.3.1) leads trivially to

$$1+z = \frac{\langle\dot{x},u\rangle_Q}{\langle\dot{x},u\rangle_P}$$

This seems to be fully generic but also rather strange - at least to me - b/c the redshift does not depend on the spacetime along C. Only the geometry at the two points P and Q is required.

It seems that between the quote above and the quote below, you realized that the quote above is not correct.

Yes, I think the generalization as described in post #41 is exactly what I was looking for. I have to find the old references (Brill, Schroedinger and Straumann, not available online) in order to understand the proof. It seems to be a fully generic formula for the redshift of a photon along its geodesic in an arbitrary spacetime. The redshift is fully encoded in the geodesic x^μ(λ) and the observer field u^μ.

That's a nice result.

It is a very nice result, and, with hindsight, a result fully expected from special relativity.

The two "facts" from the second paragraph of Wald's 5.3 are treated nicely in section 22.5 of Misner, Thorne, and Wheeler, and in the first two-and-a-half pages of section 16.2 of "An Introduction to General Relativity and Cosmology" by Plebanski and Krasinski.

 

[WannabeNewton]

As I said, Wald treats the general case

Forgive me if I interpreted your post #35 incorrectly but it seemed you were referring to a method in section 5.3 itself. I meant the method in problem 5.4, at the end of the chapter, where you show that the 4-velocity field of the isotropic observers satisfies $\nabla_{a}u_{b} = \frac{\dot{a}}{a}h_{ab}$ and then use this result when writing down how $\omega$ changes along null geodesics to derive the redshift formula given in the text.

 

[George Jones]

There is no need to look at problem 5.4 (a specific case), because the general method (not just for isotropic observers, and not just for cosmological models) is given in the second paragraph of section 5.3. This paragraph leads quickly and directly to the equation that Tom quoted.

 

[WannabeNewton]

Do you mean where he says "Thus, we can always find the observed frequency by calculating the null geodesic determined by the initial value..."?

 

[George Jones]

Do you mean where he says "Thus, we can always find the observed frequency by calculating the null geodesic determined by the initial value..."?

Yes, except that I get the reciprocal of what Tom wrote. Applying 5.3.1 twice, once at P and once at Q gives

$$1+z = \frac{\lambda_Q}{\lambda_P} = \frac{\omega_P}{\lambda_Q} = \frac{g \left(k , u \right)_P}{g \left(k , u \right)_Q}$$

There is a change of notation based on the coordinate representation of the lightlike $k$.

Assume that $s$ is an affine parametrization of the lightlike worldline and that $x^\mu$ is a coordinate system. Loosely,

$$k= \frac{d}{ds} = \frac{dx^\mu}{ds}\frac{\partial}{\partial x^\mu}$$

Denote the derivative with respect to $s$ by dot.