This is a question that has been gnawing on me for many years now. Back a long time ago, as I recall in reference to a scene in a popular science show on TV, I was asked the following.
The claim is that when you look out into the universe, you see the universe as it appeared at some past time. (This follows from the speed of light being finite, and looking at objects by observing their emitted light or other EM radiation.) The amount of time that one looks backwards is equal to the distance to the observed object in light-distance (so looking at something ten lightyears away means you are observing it as it was ten years ago).
Matter moves at a rate of speed less than the speed of light (it has to, since matter has mass, even if miniscule at the atomic level). So $v_{mass} \lt v_{EM}$ (probably significantly less, since $v_{EM} = c$).
So let's say you're looking at an object $10^{10}$ lightyears away by observing something for which $v=c$. To the observer, that object appears as it was $10^{10}$ years ago. But the Earth is much younger than $10^{10}$ years and we established that matter moves at a slower speed than EM radiation, so wouldn't the radiation that was emitted $10^{10}$ years ago long since have passed Earth's current position?
If, say, $v_{mass} = 0.5 \times c$ (a big assumption, but bear with me for a second), it would seem that the radiation emitted $10^{10}$ years ago would have passed Earth's current position $10^{10} - (0.5 \times 10^{10}) = 5 \times 10^{9}$ years ago, around the time when the solar system was still forming. Setting $v_{mass} = 0.25 \times c$ (which seems more realistic) means the radiation would have passed "us" around $7.5 \times 10^{9}$ years ago. So how could we be observing it now?
I'm not sure I'm posing this question in the best possible way (I'll admit I do find the concept somewhat confusing), and I'm sure that there's a simple explanation for it all. Just what am I missing?
I did find Is it possible to look into the beginning of the Universe? which seems peripherally related but not quite the same thing. Qmechanic brought up Seeing cosmic activity now, really means it happens millions/billions of years ago? but that question seems to be about whether it is so, not why it is so.
Answer
I think you're missing out on one of the basic results that motivated special relativity.
Light moves at the speed of light with respect to everything, regardless of the speed of that object relative to the source of the light. It doesn't matter how fast the source of the light is moving, the light emitted from the source is always going to move at the same speed light always moves at with respect to everything else in the universe: $c$.
If this sounds contradictory, it's because the equations we use for figuring out the relative speeds of things on earth don't work for speeds close to the speed of light.
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