Re: does time dilation add in series?

Dear Edward,

This is a very interesting question that deals with an important property 
of light.  We need to be careful, though, to keep in mind exactly what 
causes the time dilation or redshift.

Rather than gravitational time dilation, let me first start with special 
relativistic time dilation.  It will give us an important example of what 
happens when we return to our beginning frame of reference.  Special 
relativistic time dilation occurs when one person (we usually say a frame 
of reference) tries to measure the rate of time passing for another person 
(a different frame of reference) when the second person is moving relative 
to the first person.  The first person will measure the second person’s 
time as advancing more slowly than the first person’s, but the second 
person will also measure the first person’s time as moving more slowly, 
too!  The explanation for this seemingly impossible scenario is because of 
the way the measurements must be made.  Both individuals see the other 
person as moving and they themselves as remaining at rest.  They must 
perform similar measurements on the other person and so they arrive at 
similar conclusions.  Part of your question, though, is whether time 
dilation is additive.  The answer is in some sense yes, some sense no.  
Let’s say that Person A sees Person B as moving along the x-axis (really, 
any direction would do) and Person B sees Person C moving along the x-
axis, too, in the same direction as Person A sees Person B moving.  Person 
B will measure Person C’s time as dilated, and Person A will see Person 
C’s time as dilated more than Person B will.  The relative velocity 
between Persons A and C will not be the sum of the velocities between 
Persons A and B and Persons B and C, however!  For example, let’s say that 
Person A measures Person B as moving at 100,000km/s and Person B measures 
Person C as moving at 100,000km/s.  Person A will not measure Person C as 
moving at 200,000km/s, but rather at 180,000km/s!  This is because Person 
B measures different times and distances for Person C than Person A does.

The important point is that since velocities are not additive, the time 
dilations are not strictly additive.  The dilation can get larger if each 
successive person is moving in the same direction, but there is also 
another important point here.  If Person C measures another person, say 
Person D as moving to the left at 180,000km/s, then Person D would have no 
time dilation relative to Person A.  In other words, the time dilation 
is “reversed” when you get back to the original frame.  Basically, Person 
D would be at rest relative to Person A and so they would measure the same 
times.

Now what does all of this have to do with the redshift of light?  Well, 
you are correct that light will redshift as it leaves a star, but if we 
consider what will happen if light is gravitationally bent around a star 
something else happens, too.  As the light approaches the star, it is blue 
shifted, then the red shift as it moves away from the star returns the 
light back to its original wavelength, although it is now traveling in a 
different direction.

What about light traveling though a uniform mass distribution?  Well, the 
only way that light will be gravitationally redshifted is if it is moving 
through a non-zero gravitational field, as you mentioned, like light 
climbing up out of the gravitational field of a quasar.  A universe with a 
uniform mass distribution (dark matter for example) would not have any 
preferred direction and so would have a zero gravitational field.  Light 
traveling through such a mass distribution would not be redshifted, and 
so, a uniform distribution of dark matter would not explain the redshift 
of light from distant sources.

Well, I hope I have answered your question, Edward.  If you would like 
some more information, please let us know.

Sincerely,

Jim Guinn