Red shifting light

1. Sep 17, 2006

WendyE

Hello all,

I'm trying to figure out how light from different galaxies gets red shifted. I wonder if I could get some quick answers to a few example situations so I can get a better intuitive understanding of how light moves through space.

Q1. Say two people are moving apart at constant velocity. Person A has a flashlight that person B is observing. Does B see the light as red shiftshifted. I'm pretty sure the answer is "no" to this one.

Q2. Same two people, but they are now accelerating away from each other. Is the light red shifted for B?

Q3. The two observers are out in a portion of expanding space, but are not moving through space (relative to each other). Do they see each other moving apart? I think they will, if the space that each occupy themselves is not expanding (lets assume this, in fact). Will the light from A be redshifted for B?

If anyone can help me out with these, I'd be most appreciative.

Thanks,
Wendy

2. Sep 17, 2006

WendyE

Oh, one more question that's got me worked up:

Say two masses (A,B) are initially 1 light year apart when A starts radiating light. Space between them is expanding while the light travels from A to B. Does the light take more than 1 year to reach B? It seems like if space is just stretching, then it should take the light just 1 year to reach B, but the light will be redshifted when it gets there.

3. Sep 17, 2006

pervect

Staff Emeritus
The answer is yes, there is a redshift. WHy did you think the answer was "no"? In SR there will be a constant redshift that depends only on relative velocity and is indepndent of distance. If you are interested ultimately in cosmological redshift, there are some subtle issues in GR that make the situation a little more complex, but in special relativity in flat space-times one can equate redshift with velocity.

Yes, but the redshift factor will no longer be constant

I don't understand what you mean by "not moving" with respect to each other.

4. Sep 17, 2006

WendyE

Ok... I think I have a faulty picture in my head. Is there a distinction between moving through space, and moving because space is expanding?

5. Sep 17, 2006

Staff Emeritus

Exactly. The remote galaxies are not moving at high speed away from us through space (nor we from them). Rather the distance between us is increasing rapidly because space itself is expanding.

6. Sep 17, 2006

WendyE

Ok. Here's where my thinking is leading me. Here's a list of things I believe to be true (correct me if I'm wrong):

1. Space is expanding, and pushing galaxies apart.

2.As a galaxy gets farther away from us, the light coming from it takes longer and longer to reach us.

3. No new space is being created. The expansion is causing existing space to stretch.

Number 2 and 3 lead me to believe that it takes light longer to travel through expanded space. This is what has been confusing me. I had this idea that it takes light the same amount of time to travel through one unit of space, regardless of how expanded or contracted it is. This can't be right, though.

Have I said anything incorrect in this post?

7. Sep 17, 2006

bernhard.rothenstein

doppler effect and acceleration

Physics, abstract arxiv
physics/0609118

Radar echo, Doppler Effect and Radar detection in the uniformly accelerated reference frame

8. Sep 17, 2006

pervect

Staff Emeritus
It is standard in cosmology to assign constant coordinates, called "comoving" coordinates, to objects which move with the "Hubble flow".

Objects which are moving with the Hubble flow, though, in spite of having constant comoving coordinates, are actually increasing their distance away from each other with time.

This is handled by introducing a metric which converts changes in coordinates into distances. See for instance the wikipedia articles

http://en.wikipedia.org/wiki/Comoving_distance
http://en.wikipedia.org/wiki/Metric_expansion_of_space

The metric allows one to compute the distances between nearby points given the coordinates of those points. Because space-time is expanding, the metric is changing as a function of time.

So we have objects with constant coordinates, which are nonetheless changing their distances with respect to each other, because the metric, which converts coordinates into distances, is changing with time.

(If you are familiar with the Lorentz interval, the metric actually computes the relativistic invariant known as the Lorentz interval).

The velocity of light will be measured by any physical observer to be equal to 'c'. However, this does not mean that the velocity of light in comoving coordinates is equal to 'c'. Comoving coordinates are convenient coordinates because they don't change with time for physical bodies that move with the "Hubble flow", but the speed of light is not a constant when expressed in comvoving coordinates, it is only a constant when measured using the local clocks and rulers of a physical observer. Thus any physical observer, using his local clocks and rulers, will always measure the speed of light as 'c' - but this does not imply that the speed of light is constant in comoving coordinates, in fact a detailed analysis would show the opposite.

The path that light takes in comoving coordinates can be determined by solving the following differential equation equation

- c^2 dt^2 + a(t)^2*(dx^2+dy^2+dz^2) = 0

Here a(t) is the "scale factor" of the universe, a function which increases with time as the universe expands.

The above equation is the same as saying that light follows what is technically called a "null geodesic".

Let us suppose that y=z=constant. Then we can write

-c^2 + a(t)^2 * (dx/dt)^2 = 0

thus dx/dt = c/a(t)

and we can see that in comvoing coordinates, the rate of change of the distance coordinate with respect to the time coordinate is NOT a constant for light.

To make light travel in "straight lines", one usually replaces cosmological time t with conformal time.

For more on conformal time, try Ned Wright's cosmology tutorial
http://www.astro.ucla.edu/~wright/cosmoall.htm (some sections of which may be a bit advanced).

Last edited: Sep 17, 2006
9. Sep 17, 2006

WendyE

Wow. Thanks Prevect! That'll give me something to work on for awhile. Thanks for helping me find this information!