I Special Relativity and Time

Summary
If time slows down near the speed of light and space shrinks, doesn't that create a paradox?
So I was reading a book on special relativity and it was explaining how, if we were to go very near the speed of light, time (relative to us traveling) would slow and space itself would condense. It used the example that if we were to try and travel to a galaxy 1 million light years away, Yet we flew at 99% the speed of light it would only take us 50 years to arrive. (These are not exact numbers found using the gamma equation I'm simply paraphrasing for the concept)

However this confused me. Is this not essentially saying that "relative to us" we are travelling faster when moving SLOWER than the speed of light? Is special relativity asserting that light only takes 1 million years to reach us relative to us sitting "stationary" (not really but comparativley) on earth.

And if we were in "light's shoes" as it were, barrelling towards the earth from 1 million light years away it would really only be 50 years.

Moreover if this is correct and truly all time is relative what does that mean for our current cosmological timeline for the big bang and other historic cosmological events? How can we be sure of our measurements.

I'm sure there's something I'm missing here but I dont know what. Thank you!
 
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I was reading a book on special relativity
Which book?

It used the example that if we were to try and travel to a galaxy 1 million light years away, Yet we flew at 99% the speed of light it would only take us 50 years to arrive.
It would take us 50 years by our own clock. But we in the spaceship would also see the distance to the galaxy length contracted to less than 50 light-years.

In fact, to really describe things correctly from our frame in the spaceship, we should describe the galaxy as moving (and the Earth as well), not us. So the correct description would be that the Earth starts moving away from us at 99% of the speed of light, and at that same time in our frame, the galaxy is a little less than 50 light-years away. The galaxy moves towards us at 99% of the speed of light for 50 years.

Note that this description says nothing about how much time elapses on Earth's clocks, or the galaxy's clocks. To properly describe that you would need to take into account relativity of simultaneity--where the galaxy is "at the same time" as the spaceship and Earth are co-located depends on the frame. Failure to properly take into account relativity of simultaneity is almost always at the root of apparent "paradoxes" in SR.

Is this not essentially saying that "relative to us" we are travelling faster when moving SLOWER than the speed of light?
No. You're always at rest relative to yourself.

if this is correct and truly all time is relative what does that mean for our current cosmological timeline for the big bang and other historic cosmological events?
The times given by cosmologists are for particular observers called "comoving" observers, for whom the universe always looks homogeneous and isotropic. Observers who are moving relative to comoving observers will not observe the same times. But the times for "comoving" observers are the easiest ones to match up with the models that cosmologists use.

How can we be sure of our measurements.
Measurements are invariants and don't depend on what frame of reference you adopt. So effects like time dilation and length contraction don't make us unsure of our measurements.
 

Mister T

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Summary: If time slows down near the speed of light and space shrinks, doesn't that create a paradox?

So I was reading a book on special relativity and it was explaining how, if we were to go very near the speed of light, time (relative to us traveling) would slow and space itself would condense.
The part about time slowing is more correctly referred to as time dilation. The bit about space condensing is more correctly referred to as length contraction.

In the example you gave of traveling to a distant location we can say that according to the rest frame of the travelers the length is contracted. But according to the rest frame of Earth, time is dilated.

So you see, you don't have both length contraction and time dilation in either rest frame.

Note that a material object will always lose a race with a light beam. That is what we mean when we say material objects cannot travel at (or above) the speed of a light beam in a vacuum.
 

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