Light from distant objects was emitted soon after the Big Bang. At that time, everything was rather close together. How could it have taken so long for the light to have reached us? Instead of treating the full General Relativity problem, I will replace it with a similar problem that can be understood using Special Relativity.
Consider an explosion in the distant past (like the Big Bang) that sent the Earth and another object in opposite directions. Light emitted by the object is just now being received on Earth. This object is receding from the Earth at nearly the speed of light. Think about these events in a reference frame (Q) in which the Earth and the object have equal and opposite velocites.
Not much time passed since the explosion in the rest frame of the emitting object, but a lot of time passed in the Earth frame. This is illustrated in the spacetime diagram below, where (x,t) are the coordinates of frame Q; (x', t') are the coordinates of Earth. The explosion is at the origin for both systems.
In this analogy, the rate of expansion of the universe is assumed constant. General Relativity effects are not properly treated, but the rigorous solution would be qualitatively similar.
This is related to the famous Twin Paradox, in which the traveling twin ages much less than the stay-at-home. As with the Twin Paradox, the resolution lies in the relativity of simultaneity. In the Earth frame, the emission took place much later in the evolution of the universe, whereas in the rest frame of the object, it is nearly simultaneous with the Big Bang.
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