Our expanding universe is not thought to be relativistic. Technically, the galaxies receding away from us are at rest. The space between them is expanding and creating the impression that the galaxies are in motion. However, each galaxy can be treated as a clock, and, it is not likely that any of these clocks tell time at the same rate.
Using the diagram below as a reference, consider Alice and Bob living in a static universe. Alice lives in galaxy A and Bob lives in galaxy B. Each has a photon gun and shoots one photon per second at the other. The vertical red arrows represent the photons fired from one galaxy to the other. When the first photon from Alice (after traveling light years) finally arrives at galaxy B, Bob intercepts it and records it as a unit of time. Since Alice fired photons at regular intervals, Bob only has to wait one second for the next photon and another second for the next one after that and so forth. He receives one photon per second from Alice, and, for similar reasons, Alice receives one photon per second from Bob. This is how Alice and Bob keep track of each other's time rate.
You will note there are some horizontal red arrows at the top of the above diagram. Also located at galaxy A is Gertrude. She also has a photon gun. She fires one photon per second at Alice. Alice therefore intercepts and records one photon per second from Gertrude. Bob also receives photons in the same manner from Norbert who is also lacated at galaxy B (see horizontal red arrows at the diagram's bottom). This is how Alice and Bob keep track of their own time rates.
When the universe is static, both Alice and Bob receive fired photons from all directions at a rate of one photon per second. All clocks seem to agree and equations 1 and 2 provide the relevant math. But suppose the galaxies move away from each other at velocity v due to expanding space. The next diagram represents this scenario:
The vertical red arrows show that the photons take longer to reach their respective destinations and fewer photons are received per second compared to photons represented by the horizontal arrows. Since each photon is counted as a unit of time, Alice and Bob have the impression that the other's time rate is slower. Since Gertrude is located in the same galaxy as Alice, she isn't receding from Alice the way Bob is, so Alice sees no change in her time rate, since she receives the same number of photons (time units) from Gertrude. Ditto for Bob and Norbert at galaxy B.
If we do a little algebra we can derive the Lorentz equation from equation 3 above. The final result is equation 10:
Equation 10 shows each galaxy's proper time (t') shrinking as they accelerate further apart. It does not matter if the galaxies are technically at rest, since the space inbetween and the photons are not at rest. If the space is at rest, here's the result after firing a photon gun for five seconds:
Notice the steady stream of photons. This is where t' = t. The next diagram shows what happens if space expands at a steady rate of v. The photons still have a steady rate but the interval between them has increased, so they are not counted as frequently. Time t' is less than t.
Finally, the next diagram is the most realistic, since the expansion of space is accelerating. Here the the interval between photons continues to grow and they are counted less and less frequently. Time t' is shrinking.
When Alice and Bob are far enough apart, neither will receive any photons from the other. Time t' will be zero. Thus the expanding universe is relativistic if one keeps track of the various time rates at different distances.
