Why aren't we all moving at the speed of light relative to photons?

Question

Using basic logic.... We know speed and velocity is relative. My point is if we take our speed with respect to a photon moving in opposite direction we are practically moving at speed greater than that of light...... Is my argument correct?

Answer 1

No your argument is not correct. Firstly, velocities do not add linearly like 3-vectors in Euclidean space: the relativistic sum of two velocities always has a speed of less than $c$ if both the velocities' magnitudes are less than $c$ (no matter what their direction).

Secondly, photons have no rest frame: that's a basic property of things that have zero rest mass. A's speed relative to B is the speed of A measured from the frame wherein B is at rest. So let $B$ be your photon and $A$ be some other observer. But in this case, B has no rest frame. So we cannot give a meaningful answer to what $A$'s velocity relative to $B$ is. If $B$ were something other than a zero rest mass object and were moving relative to us at $c-\epsilon$ where $\epsilon$ were as small as you like, then we wouldn't run into this problem and you could calculate the relative velocity. But, as stated above, this relativistic sum cannot have a magnitude greater than $c$.

So if your question were, do we move at velocities near $c$ relative to, say, an electron accelerated to near the speed of light relative to us, then the answer would be "yes we would move at that same near to $c$ speed relative to the electron".