Galaxies travelling away from each other faster than light

[Tombo254]

This might seem simple to some of you on here, but I have a question that's been bothering me, most likely due to my lack of understanding, not an actual problem. If two Galaxies, A and B, are traveling away from each other, say each at 3/4 the speed of light, does that not mean that if you are on galaxy A, you will see galaxy B traveling away from you at 3/2c?

 

[Sciurus]

If two Galaxies, A and B, are traveling away from each other, say each at 3/4 the speed of light, does that not mean that if you are on galaxy A, you will see galaxy B traveling away from you at 3/2c?

Good question. But, no, they would appear to move apart at 0.96c from one galaxy looking at the other, if I worked the problem right. It's one of the funny effects of special relativity (http://en.wikipedia.org/wiki/Introduction_to_special_relativity).

 

[Tombo254]

how do you calculate that?

 

[bcrowell]

If it was special relativity, you'd calculate it using the relativistic velocity addition equation: http://en.wikipedia.org/wiki/Velocity_addition#Special_theory_of_relativity The result would come out to be less than c.

Since this is general relativity, the SR equation doesn't apply. Here is a FAQ entry that may be helpful.

FAQ: What does general relativity say about the relative velocities of objects that are far away from one another?

Nothing. General relativity doesn't provide a uniquely defined way of measuring the velocity of objects that are far away from one another. For example, there is no well defined value for the velocity of one galaxy relative to another at cosmological distances. You can say it's some big number, but it's equally valid to say that they're both at rest, and the space between them is expanding. Neither verbal description is preferred over the other in GR. Only local velocities are uniquely defined in GR, not global ones.

Confusion on this point is at the root of many other problems in understanding GR:

Question: How can distant galaxies be moving away from us at more than the speed of light?

Answer: They don't have any well-defined velocity relative to us. The relativistic speed limit of c is a local one, not a global one, precisely because velocity isn't globally well defined.

Question: Does the edge of the observable universe occur at the place where the Hubble velocity relative to us equals c, so that the redshift approaches infinity?

Answer: No, because that velocity isn't uniquely defined. For one fairly popular definition of the velocity (based on distances measured by rulers at rest with respect to the Hubble flow), we can actually observe galaxies that are moving away from us at >c, and that always have been moving away from us at >c.[Davis 2004]

Question: A distant galaxy is moving away from us at 99% of the speed of light. That means it has a huge amount of kinetic energy, which is equivalent to a huge amount of mass. Does that mean that its gravitational attraction to our own galaxy is greatly enhanced?

Answer: No, because we could equally well describe it as being at rest relative to us. In addition, general relativity doesn't describe gravity as a force, it describes it as curvature of spacetime.

Question: How do I apply a Lorentz transformation in general relativity?

Answer: General relativity doesn't have global Lorentz transformations, and one way to see that it can't have them is that such a transformation would involve the relative velocities of distant objects. Such velocities are not uniquely defined.

Question: How much of a cosmological redshift is kinematic, and how much is gravitational?

Answer: The amount of kinematic redshift depends on the distant galaxy's velocity relative to us. That velocity isn't uniquely well defined, so you can say that the redshift is 100% kinematic, 100% gravitational, or anything in between.

Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf

 

[Sciurus]

The link is the formula for a projectile fired from a moving platform, which is mathematically similar to what you are asking; I used the "velocity addition" formula of special relativity.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel2.html

 

[Sciurus]

Thanks, bcrowell.

Sorry, Tombo254. I assumed special relativity applied everywhere.