# How relative is speed?

A short question, because the math got me confused. If in my inertial reference frame, I see a dude going away from me at a speed $$v$$. Then, from his frame of reference (which is also inertial) he also sees me moving with speed $$v$$. No matter how close he is to the speed of light.

Is that correct ? $$dx/dt = v = dx'/dt'$$ (the gammas cancel out)(I chose $$x'$$ to be opposite in direction to $$x$$)

• Commented Aug 6, 2020 at 15:32

It sounds right, as no dude has a preferred frame of reference. To be sure, let's use the infallible Lorentz transformation (though I cannot, in good conscience, invert $$x'$$):

Suppose you remain at:

$$E_1 = (t_1, x_1) = (0, 0),$$

while the dude starts there. After a time $$T$$, he has moved to $$x=D$$, placing that event at:

$$E_2 = (t_2, x_2) = (T, D)$$

while you remain at:

$$E_3 = (T, 0)$$

for which you measure a speed:

$$v \equiv \frac{x_2-x_1}{t_2-t_1}=\frac D T$$

Transforming to his frame using:

$$t' = \gamma(t-\frac{vx}{c^2})$$

$$x' = \gamma(x-vt)$$

dude sees:

$$E_1 = (\gamma(0-0), \gamma(0-0))' = (0, 0)'$$

$$E_2 = (\gamma[T-\frac{vD}{c^2}],\gamma[D-vT])'$$ $$E_2 = (\gamma[T-\frac{D^2}{Tc^2}],\gamma[D-DT/T])'$$ $$E_2 = (\gamma[T-\frac{D^2}{Tc^2}], 0)'=(\gamma[T(1-\frac{v^2}{c^2})], 0 )' = (T/\gamma, 0)'$$

meaning he didn't move and his clock ticked slower. Meanwhile, you moved to:

$$E_3 = (\gamma(T-0), \gamma(0-vT))' = (\gamma T, -\gamma D)'$$

for a velocity of:

$$v' \equiv \frac{x'_3-x'_0}{t'_3-t'_0}=\frac{-\gamma D}{\gamma T} = -v$$

Note that in your frame, each participant measures the other's speed at the same time, while in dude's frame, you measured his speed before he measured yours.

• thanks a lot ! I love your usage of "dude" Commented Aug 3, 2020 at 13:55

Yes you are absolutely correct, if you see someone travelling at a speed $$v$$ (be careful on the distinction between speed and velocity), they also see you travelling at a speed $$v$$, regardless of how close you are to the speed of light (but not travelling at the speed of light).