# How does time dilation impact the observed speed of light?

so I've been struggling somewhat with rationalising this concept, so I'll explain my thinking and hopefully someone can point out the flaw.

My initial reading had been that, if an observer is moving towards a light source in the hopes of adding their speed to the perceived velocity of the incoming light, that same velocity will slow the observer's time by an equal amount - their measurement devices' slowing accounting for the difference.

But surely that would actually make the light appear to arrive much faster, less time having elapsed for the observer relative to, say, a second static observer.

Similarly, I have a hard time working out precisely how any time dilation effect can make the 'right' correction accounting for either approaching or retreating from a light source.

I really hate struggling so badly over a relatively (.....heh) simple principle, so I'd greatly appreciate your clarifications. If you're able to put it in similar terms to those I've used that would also help a great deal. Thanks.

• I think what you are trying to describe is the relativistic doppler effect, rather than the addition of velocities - but it's not really clear to me what the question is. Apr 13 '17 at 1:11
• I think you should consider space as well and then see if this clarifies how light is able to be perceived the same regardless of reference frame. Apr 13 '17 at 1:35

You are failing to account for the fact(s) that when you start to move, you revise your opinion about all sorts of things, including your distance to the source of the light beam and the time when the light beam left that source.

Instead of one person who starts to move, let's compare two people, one of whom is already moving relative to the other.

Alice and Bob are standing at the same place at the same time, observing the same light beam, with both their watches set to show time $0$. Alice considers herself standing still, while Bob (according to Alice) is moving toward the light beam at speed $3/5$.

Here is Alice's diary (with all times measured in minutes):

Time 0: Bob is right next to me, moving eastward. Jeter, who is 8 light-minutes east of here, just turned on his flashlight.

Time 5: In the past 5 minutes, Bob has traveled 3 light-minutes eastward, and has just been hit by Jeter's light beam. The beam started out 8 light-minutes eastward, so it must have traveled 5 light-minutes. Its speed is therefore 1 light-minute per minute.

Here is Bob's diary:

Time -6: Jeter, who is 10 light-minutes east of here, has just turned on his flashlight. (Alice is between us, 3.6 light-minutes to the east.) [Note that the turning on of the flashlight is the same event that, according to Alice, takes place at time 0.]

Time 0: Here I am, crossing paths with Alice. It took 6 minutes for her to cover that 3.6 light-minutes. Sure enough, our relative speed is 3/5 light-minute per minute.

Time 4: And here is Jeter's light beam, hitting me. It took 10 minutes to cover the 10 light-minutes between us. Sure enough, the speed of that light is 1 light-minute per minute. [Note that this is the same event that, according to Alice, takes place at time 5.]

When you start to move toward the light source, you turn yourself from Alice into Bob, and revise all your calculations accordingly. But the new calculations, like the old ones, will give you a light-speed of one.

You are correct that the observer that was initially at rest will see that the clock of the moving person runs slower. The moving observer will also see that the distance between the light source and himself contracted in the same proportion. Thus, he will measure a smaller time, because to him the light had to run a smaller distance. But the ratio, that is the speed, remains the same.

In your system of reference, the length has not contracted. But you justify the smaller time measured by the observer as an artifact due to clock desynchronization. That is, to you, the clock of the observer and the clock at the place of light emission, will be desynchronized. If you had them synchronized, then they will be not synchronized for the moving observer.

Imagine that each point in spacetime has two clocks, one at rest with you and another moving with respect to you. If you look at different positions, only one of the pairs can be in synchrony, the rest will not (I mean, distances along/parallel the line of motion).