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Let’s say an observer is travelling at 5% the speed of light. In the opposite direction a projectile approaches the observer at 99.9% light speed. According to the formulas for time dilation and length contraction, this is almost negligeable at 5% light speed. Since this is the case, shouldn’t the observer see the projectile going almost 105% the speed of light? Of course the observer will see it go slightly below light speed, but why? What physical phenomenon is causing the observer (who is hardly experiencing any time dilation or length contraction) to still see the projectile going no faster than light speed?

Luppo_Alberts
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    Let A be the observer measuring the 5% and the 99.9% and B be the observer in your question, going 5% of $c$ relative to A. The fact that 5% is "small" just means that A and B measure nearly the same speeds of the projectile. In the limit where the 5% goes to 0%, A and B must obviously agree on what the speed of the projectile is. – Marius Ladegård Meyer Apr 24 '23 at 21:34
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    Re, "a projectile approaches the observer at 99.9% light speed." I think most readers would understand that sentence to mean that the .999C velocity was relative to the observer. But I think you actually are talking about two observers. I think you are saying that Observer A sees Observer B moving to the left at .05C, and she simultaneously sees the projectile moving to the right at .999C. Given those facts, I think you are asking what Observer B sees. The speed of the projectile, as seen by Observer B, is given by the formula in Dale's answer: It works out to approximately 0.999095C. – Solomon Slow Apr 25 '23 at 00:15
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    Re, "What physical phenomenon is causing the observer...to still see...?" I don't think that "physical phenomenon" is the right way to describe it. The fact that observer B sees the projectile moving at less than the speed of light IS the physical phenomenon. The theory that predicts it is the Special Theory of Relativity. That theory is a mathematical consequence of two things; (1) that the speed of light is independent of any motion of the light source relative to the measuring instrument, and (2) our desire for the laws of physics to be the same in every inertial coordinate system. – Solomon Slow Apr 25 '23 at 00:28
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    Curvature of space time is general relativity, not special. Are you asking how special relativity relates to general relativity, or do you want an analysis of this problem only in terms of special relativity? In the latter theory, no observer experiences curvature of space time, only linear contraction/dilation. – phoog Apr 25 '23 at 11:07

2 Answers2

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According to the formulas for time dilation and length contraction, this is almost negligeable at 5% light speed.

While that is true it is not relevant. What you are discussing with the projectile is not time dilation or length contraction, but rather velocity addition. The relativistic velocity addition formula is $$\frac{u+v}{1+uv/c^2}$$ We can do a second-order Taylor series expansion to get $$\frac{u+v}{1+uv/c^2}=u+v-\frac{u^2 v}{c^2} - \frac{u v^2}{c^2} + O\left(\frac{v^3}{c^3},\frac{u^3}{c^3}\right)$$ In this expansion, even if $u\approx 0$ if $v\approx c$ then the relativistic velocity addition formula is not well approximated by the Galilean $u+v$.

Dale
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    Might be worth adding that for u=0.999000000c and v=0.050000000c, the formula yields the result 0.999095195c. (I'm showing far more precision than would be appropriate, just so the curious can see the effect.) – Ben Hocking Apr 25 '23 at 14:07
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Let's say projectile speed measuring device consists of two clocks and a ruler.

After changing the speed of this device by 0.05 c, it is necessary to do a adjustment of the clocks.

This adjustment is a large one. It is large enough to correct the 5% error that occurs without the adjustment. Here 'error' means the difference of measured speed and the 'correct' speed.

stuffu
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