Light Bulb Moving at Half the Speed of Light: Effects on Light Halo and Length

[dragoneyes001]

quick question when a light source is moving at a good clip like the above where the light bulb is moving at half the speed of light am I wrong in the drawing that the light halo would be lagging towards the back but even on the two sides? what I'm wondering is how long would the light behind the bulb be? the same length plus the speed at which the bulb is moving or simply the exact length of the light emitted as on the sides ?

 

[BiGyElLoWhAt]

I think you're viewing the light as a ray instead of discrete packets. The light to the leftof the bulb would be red shifted (have a longer wavelength) while the light on the right would be more blue shifted (have a shorter wavelength), but both waves (left and right) would be propagating at the same speed. If you were wondering how much the shift would be, it should be as simple as some pythagorean's theorem mixed with some basic wave kinematics. SR is relatively (ha!) simple, mathematically.
Start with the bulb at one position, measure the distance from the bulb to one peak of a wavelength, then allow the bulb to move a certain distance as well as the light to propagate for the same amount of time that it took the bulb to move that certain distance, and calculate the distance between the second peak (the one most recently emitted by the bulb) and the previous peak. The ratio or delta, whichever you're looking for, is the amount of red/blue shift of the light.

 

[Orodruin]

The drawing is conceptually correct. The bulb will be further away from the wave front emitted in the backward direction than that emitted in the forward direction. As BiG said, it will be red/blue shifted, but that is not really of importance for the movement of the wave front.

 

[Nugatory]

quick question when a light source is moving at a good clip like the above where the light bulb is moving at half the speed of light am I wrong in the drawing that the light halo would be lagging towards the back but even on the two sides? what I'm wondering is how long would the light behind the bulb be? the same length plus the speed at which the bulb is moving or simply the exact length of the light emitted as on the sides ?

The speed of light is the same for all observers. An observer moving along with the light source will find himself at the center of a spherical shell.

An observer moving to relative to the light source will also find the light emitted at any moment to be expanding in a spherical shell; however the light source will move away from the center of that shell as it expands outwards. The observer moving relative to the light source will also observe red shifts and blue shifts; whether it's the left-moving or the right-moving light that is red-shifted depends on whether this observer is moving to the left or to the right relative to the source.

And what about an observer at rest while the source is moving? That's the exact same situation as the source at rest and the observer moving in the opposite direction.

 

[BiGyElLoWhAt]

@Orodruin , since I'm still also new to this matter, let me ask a follow up question. How would the wavefronts appear if I were sitting on the bulb as it moved at .5c? If it were still to appear the same (I'm pretty sure length contraction only affects length components perpendicular to the velocity) as in the diagram, wouldn't that contradict the premise of SR? The photons to the left would appear to move above c, while the components to the right would appear to move slower than c.

Edit* Since Nugatory invoked the question I was asking, let me tag him in this as well @Nugatory

 

[BiGyElLoWhAt]

Retracted. A quick google search set me straight.

 

[Orodruin]

Even if you can google it, let me answer it for completeness.

An observer on the bulb will see the bulb moving at zero velocity and the light at c in all directions. Therefore, the wave front will be spherically symmetric around the bulb as seen by this observer.

Edit: Nugatory essentially already said this. I should stop reading threads backwards ...

 

[BiGyElLoWhAt]

Yes, that is a statement of one of the primary postulates of SR, but not really an explanation, honestly. The answer is that the x component length contraction is caused by x component velocity, as y with y, and z with z. I just don't work with SR on any kind of regular basis, and was completely upside down. It's parallel components that contract, not perpendicular components.

 

[Orodruin]

Yes, that is a statement of one of the primary postulates of SR, but not really an explanation, honestly. The answer is that the x component length contraction is caused by x component velocity, as y with y, and z with z. I just don't work with SR on any kind of regular basis, and was completely upside down. It's parallel components that contract, not perpendicular components.

Be warned that you cannot apply length contraction to this in an easy fashion. Length contraction is based on a moving object with a fixed rest length, which the distance between the bulb and the wave front is not.

 

[BiGyElLoWhAt]

Hmm... Can you not look at one instance in time, knowing the velocity of the bulb relative to whichever observer it's moving at .5c w.r.t.? Then, in this instant, you have one length for each side, and you should be able to apply it, at least in my mind. Since dilation is a ratio, it will keep it the same for all instances in time. Of course, I'm essentially separating space and time, but most examples demonstrate spacetime as one entity by assuming space and time are separate entities, then showing how one affects the other, and vice versa.

 

[Orodruin]

No, the length contraction formula is based on having an object with fixed rest length and then using the fact that it is not moving in that frame. In this case, the "object" (wave front to bulb distance) is growing with time in any frame.

 

[Nugatory]

Yes, that is a statement of one of the primary postulates of SR, but not really an explanation, honestly. The answer is that the x component length contraction is caused by x component velocity, as y with y, and z with z. I just don't work with SR on any kind of regular basis, and was completely upside down. It's parallel components that contract, not perpendicular components.

There is no length contraction anywhere in this situation, only redshift and blueshift for the observer moving relative to the source.

Now you may be wondering how the moving and the stationary observers can both find themselves at the center of a spherical shell of light expanding outwards in all directions at speed $c$... When we say that the expanding wavefront is a sphere, we're saying that the wavefront in all directions is at the same distance from the source at the same time. And "at the same time" is, thanks to the relativity of simultaneity, different for the different observers.

 

[dragoneyes001]

my example was of a stationary observer far enough away to be able to observe a light source wiz by. from the answers I'm seeing the observer wouldn't be able to distinguish any observable change to the light halo's shape in relation to the moving object correct?

 

[BiGyElLoWhAt]

No, the length contraction formula is based on having an object with fixed rest length and then using the fact that it is not moving in that frame. In this case, the "object" (wave front to bulb distance) is growing with time in any frame.

Ok, but even if we were to have an object which had a growing length with respect to time along some axis, I don't see how that would change gamma.

There is no length contraction anywhere in this situation, only redshift and blueshift for the observer moving relative to the source.

Now you may be wondering how the moving and the stationary observers can both find themselves at the center of a spherical shell of light expanding outwards in all directions at speed $c$... When we say that the expanding wavefront is a sphere, we're saying that the wavefront in all directions is at the same distance from the source at the same time. And "at the same time" is, thanks to the relativity of simultaneity, different for the different observers.

Hmm... I'm not going to ask you to walk me through this, but I will say that I don't see how time dilation explains it. Light always moves at c, and the only way I can see for our observer to witness the distance, or length, between the wave fronts is by seeing one of the distances to dilate by one amount and the other to dilate by a different amount. In order for me to see time dilation as an explanation, our observer must have time remain stagnant for themselves, that way their position never changes w.r.t. the wave front. I will however work through it and see what comes from it.

my example was of a stationary observer far enough away to be able to observe a light source wiz by. from the answers I'm seeing the observer wouldn't be able to distinguish any observable change to the light halo's shape in relation to the moving object correct?

if the observer is stationary w.r.t. the center of the halo, there will be no observable change to the shape, nor the wavelength. According to ordruin and nugatory, an observer in motion w.r.t. the center of the halo will observe a red/blue shift, but not a change in shape. So the short answer is presumably yes.

 

[Orodruin]

Ok, but even if we were to have an object which had a growing length with respect to time along some axis, I don't see how that would change gamma.

It does not, but a growing object has at least one end which is not at rest and the definition of rest frame for such an object is ambiguous. The easier way of looking at it is to realize that the gamma appears as a result of assuming constant rest length. If one end of the object starts moving you are further going to have additional problems of simultaneity in the rest frame vs moving frame (whatever you define the rest frame to be).

Hmm... I'm not going to ask you to walk me through this, but I will say that I don't see how time dilation explains it. Light always moves at c, and the only way I can see for our observer to witness the distance, or length, between the wave fronts is by seeing one of the distances to dilate by one amount and the other to dilate by a different amount. In order for me to see time dilation as an explanation, our observer must have time remain stagnant for themselves, that way their position never changes w.r.t. the wave front. I will however work through it and see what comes from it.

There was no reference to time dilation here. Only a reference to the ambiguity of the word simultaneous as a logical consequence of SR is that what is simultaneous in one frame is not necessarily simultaneous in another.

 

[BiGyElLoWhAt]

There was no reference to time dilation here. Only a reference to the ambiguity of the word simultaneous as a logical consequence of SR is that what is simultaneous in one frame is not necessarily simultaneous in another.

Is this not an indirect reference to time dilation between 2 observers? Of course referencing some coordinated t=0 between the 2 observers.

... And "at the same time" is, thanks to the relativity of simultaneity, different for the different observers.

The point I was making, was that regardless of our choice of a second observer, our observer moving at the same velocity w.r.t. the light source could only see the distances between themselves and the wave front on either side (parallel/antiparallel to the velocity w.r.t. our 2nd observer) as the same is if those distances appeared differently between themselves and our 2nd observer. I still am not seeing how that is not length/spacial coordinate dilation.

 

[Orodruin]

Is this not an indirect reference to time dilation between 2 observers? Of course referencing some coordinated t=0 between the 2 observers.

I would say the inference is the other way around. The relativity of simultaneity implies time dilation. It is not a priori the same thing. Depending on spatial position of the events, observers may disagree on which order the events occur in (assuming space like separation of the events).

 

[ghwellsjr]

quick question when a light source is moving at a good clip like the above where the light bulb is moving at half the speed of light am I wrong in the drawing that the light halo would be lagging towards the back but even on the two sides? what I'm wondering is how long would the light behind the bulb be? the same length plus the speed at which the bulb is moving or simply the exact length of the light emitted as on the sides ?

As has been pointed out by others in this thread, in the frame in which the light bulb is moving as you show in your picture, the expanding ring of light is a perfect circle centered on the location where the bulb was when the flash was set off. But it is also a perfect circle in the frame in which the light bulb is at rest. However, what has not been pointed out is the fact that observers cannot see these rings of light or determine that they form perfect circles. They can only see light that is at their locations.

So if we assume that both observers are collocated with the light bulb when the flash occurs but one of them remains at rest in the frame depicted above while the other one moves with the light bulb, the only way that they can establish that they are each in the center of the ring of light is for each of them to put a ring of reflectors around themselves and wait until the light reflects back to them to see if it arrives simultaneously from all reflectors.

I have made an animation to depict this scenario. Please press the play button in the lower left, not the one in the center:



As you can see, the mirrors are spaced in a perfect circle for the green stationary observer but it is an ellipse for the red moving observer. This is because the red observer is moving in this frame and his measured distances along the direction of motion are subject to Length Contraction although as far as he can tell, the mirrors are all an equal distance from him.

You will note also that the red moving observer is subject to Time Dilation as it takes longer for him to see the reflections than it does for the green stationary observer, but again, he is not aware of this because any clock he carries with him is subject to the same amount of Time Dilation so both observers will establish that the light takes the same amount of time to make its round trip, assuming they both placed their mirrors the same distance apart.

Another thing to note is that the light forms perfect circular rings or partial rings as it expands outward and as the reflections collapse inward. However, all the reflections happen simultaneously for the green stationary observer but at different times for the red moving observer, although he cannot tell that they are not happening simultaneously.

 

[dragoneyes001]

thank you that animation actually answered a few other things for me too.