Doppler Effect and Relativity paradox

Question

Let, Alice is moving towards Bob at very high speed. Therefore, events in Bob's frame will appear to happen slowly in Alice's frame due to time dilation. Since velocity is relative, the same is also true for Bob.

Now, since Alice is moving towards Bob, Bob's light will be blueshifted towards him. Special relativity will cause it to blueshift even greater. That means, time between two light pulses, therefore two events on Bob's actually decreases as seen by Alice. Not to mention, the same thing will also happen for Bob.

So Alice sees the events at faster than normal speed even though they are happening at slower than normal speed. How is this patadox resolved?

Answer 1

"That means, time between two light pulses, therefore two events on Bob's actually decreases as seen by Alice."

No.

c=λf=λ/Τ

c in vaccum space must be fixed at all times. The frequency will increase (blueshif) but the wavelength will proportionally decrease to keep c fixed. Meaning the distance between B and A will contract to compensate for the faster pace of time.

A blinking white light for example with a 1 sec pause period will be measured for both B and A to blink at the same rate with the only difference that both B and A will see it now blueshifted in color.

You confuse SR time dilation and length contraction with some kind slow motion video effect. This is a common misconception of SR.

What SR says about your light perception (image) of a relative moving light source towards a target observer, is at the time light has reached you the only effect you will observe will be a color change (Doppler shift). This is now because the increased momentum thus energy of a moving light source does not affect the propagation speed of light which remains fixed but only its frequency E=hf.

Answer 2

The Doppler effect and time dilation are two separate phenomena- the first will either reinforce or counteract the apparent effect of the other, depending on the direction of the relative motion concerned.

The Doppler effect is an apparent change in clock rate that results from changes to the distance light has to travel to reach an observer. The clock rate appears to increase when the distance is reducing (ie when the observer is moving towards the source), and to decrease when the distance is increasing.

Time dilation is a quite separate effect which is caused by the geometry of spacetime, and reflects the fact that different paths between two events can have different durations. If the path you follow between two events lasts for 10 minutes, while the path I follow between the same events lasts for 11 minutes, the elapsed time on you clock will be less than the elapsed time on mine- that is not because your clock has 'run slow', but because it has taken a shorter path through time.

If a clock is moving towards you, the Doppler effect will cause it to appear to speed up, thus countering the effect of time dilation. If the clock is moving away from you, the Doppler effect will cause the clock to appear to slow, thus adding to the effect of time dilation.