Question

How does the Doppler Effect prove that light travels at a constant speed and is not affected by vector addition or by an object moving to and fro?

Please give a very detailed reason. Thank you.

Answer 1

We can take the relativistic doppler effect to understand the speed of light and its addition to any other velocity

According to the relativistic addition of velocities let us say "u" and "v" assuming they are very close to the speed of light or say they are relativistic in nature.

Then

so here u and v are two velocities which are to be added using doppler effect principle, c is the speed of light and u' is the resultant velocity of u and v

now let us say u=c (one of the velocity is equal to velocity of light)

case-1 assume a source of light is travelling on a train which is moving at a velocity v, so the light coming from the source is travelling at a speed of c and for somebody observing the train, the light's velocity should get added to the velocity of train

if we use the formula,

that means the velocity of light will not get added to any velocity. This can be used with different variations of the formula let us say u+v or u-v

now a very interesting example, is let us try adding c to c

then

So speed of light is the ultimate quantity which can never be added to anything or subtracted from anything

there are many more interesting examples let say 0.99c and 0.99c added vectorially will still be smaller than c

Hope this helps. Cheers.