# Hypothetical Brain Teaser about Relative motion and the finite speed of light

The essay question I have chosen for my assignment is the following:

" If you are traveling in a car at the speed of light and you turn on your headlamps, will the light emitted illuminate the path in front of you?"

I am not concerned with the car or its occupants in any way. The effect of relative motion at the speed of C on white light is more the point of this essay. In an effort to create a well rounded paper, I am trying to use the Reference frames of; motionless & behind the car, motionless & in front of the car and inside of the car looking forward. I have tried to use the Relativistic Doppler Shift Formula to calculate these. My attempt at calculating the blue shift yielded a value of -1.59578..... for Z. I had expected to get an answer of 1 or -1. Which, I think, would mean a relative motion of hundred percent of the speed of light.

"Is it still accurate to use the Relativistic Doppler Shift while using C for the velocity?"

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Andrew Mason
Homework Helper
The essay question I have chosen for my assignment is the following:

" If you are traveling in a car at the speed of light and you turn on your headlamps, will the light emitted illuminate the path in front of you?"

I am not concerned with the car or its occupants in any way. The effect of relative motion at the speed of C on white light is more the point of this essay. In an effort to create a well rounded paper, I am trying to use the Reference frames of; motionless & behind the car, motionless & in front of the car and inside of the car looking forward. I have tried to use the Relativistic Doppler Shift Formula to calculate these. My attempt at calculating the blue shift yielded a value of -1.59578..... for Z. I had expected to get an answer of 1 or -1. Which, I think, would mean a relative motion of hundred percent of the speed of light.

"Is it still accurate to use the Relativistic Doppler Shift while using C for the velocity?"
The car cannot travel at the speed of light. It can, theoretically, get arbitrarily close to c but it cannot reach it. Light will always move away from the car at the speed of light.

The headlights look the same regardless of your speed. In fact, relative to distant parts of the universe, earth cars are traveling at very close to c. If you were to accelerate from earth's frame and travel at a speed close to c, the rest of the universe that may look a little different, and light reflecting off objects as you approached them would be doppler shifted to a higher frequency, but the light always moves away from the car at the speed of light.

AM

So even by imparting a tremendous amount of relative speed upon light between 400 and 750 nm you could not change it from one "type" of light to another? It seems to me that the variation in appearance of the light from someone watching headlights drive away at C and someone watching headlights come toward them at C would be great enough to change the appearance of the light. The idea that C is not as much a speed limit as a concrete fact/necessity I understand, but it also seems to me that their should be a way to calculate shift in frequency when giving white light the added relative motion of approximately 300,000,000 m/s. Are these ideas false?

diazona
Homework Helper
You can change light from one "type" (frequency) to another even by imparting a small amount of speed to the source. But the formulas for this effect are not really well defined for velocity equal to c. If you look at the relativistic Doppler shift formula (from Wikipedia: http://en.wikipedia.org/wiki/Relativistic_Doppler_shift), you can see that plugging in c for the velocity gives a "Doppler factor" of

sqrt( (1-c/c)/(1+c/c) ) = 0

So if you could somehow see a car zooming past you at the velocity of light with its headlights on, the light would have a frequency of 0. But light at frequency 0 carries no energy, so in a sense it doesn't really exist at all.

Or in the other direction,

sqrt( (1+c/c)/(1-c/c) ) = infinity

which means the light would have an infinite frequency and infinite energy, which gets into all sorts of weirdness (photons collapsing into black holes and so on). But of course it can't happen, since no matter can move at the speed of light, so no matter how weird the answer is we don't really have to worry about it.

I think it is important to remember that, I think, It is not that nothing can break the speed of light barrier. Rather that it is information that cannot travel faster than the speed of light. Which, I also think , makes breaking the speed barrier somewhat of a moot point seeing as they're is no point of reference or way to receive information from something breaking the light speed barrier, but it is not a concrete impossibility(save things that have mass) which light does not. I also believe the infinity answer is incorrect. Having zero as a denominator makes the equation undefined but remembering that sqrt((1+c/c)/(1-c/c)) -1 is equivalent to z. so adjustments can to made to avoid making the equation undefined. I hope that my responses are not seen as overly argumentative but simply my effort to find an answer as well as contribute to the discussion. thanks :)

diazona
Homework Helper
Yes it is a concrete impossibility, if relativity is correct (and every indication is that it is). Massive particles can only move at speeds less than c, and massless particles (like photons, i.e. light) can only move at the speed of light. Admittedly it can sometimes be misleading to say that "nothing travels faster than light" - for instance, the phase velocity of a quantum mechanical wavefunction can be greater than c, if I recall correctly - but that doesn't correspond to anything physical.

As for your assertion that the infinity answer is incorrect: have I made a mistake? Or do you doubt the validity of basic math? I'm curious what kinds of adjustments you think you can make to prevent that expression from being undefined. (I also hope I'm not coming across as overly argumentative, it's only because I think you need to justify the statements you've made)

You must ask the question more clearly. What do you really mean when you say your car is travelling at the speed of light? As measured how? In what dimension?

Are you attempting to use an observer? Where is that observer located? Is the observer's frame also the car's frame?

Most often, this particular question is meaningless. Usually people who ask this are talking about a physically impossible situation.

The impossibility is what I was getting at I suppose, with the majority of the question. I was curious if any one knew of a form of wavelength frequency( which is what these all are) where I can get a concrete number for the output. If one were to calculate the speed of a projectile then the answer for the speed of light leaving headlights traveling at the speed of light is c or 1 which makes sense and when running the numbers for the car and virtually anything else in this question then yes it is an "impossible situation" but I can not think any reason why you can not figure out what the Doppler shift would be of light if you input the relativistic speed of it's source as light. As I think about it, when you give c the v of c this does not mean you have to break the speed of light and violate theory. So can someone explain to me why you can't or if it is in any way mathematically possible to figure this. I asked the question broadly to try and open it to any other unique(means I did not see it) suggestions or p.o.v. on this type of situation

diazona
Homework Helper
The impossibility is what I was getting at I suppose, with the majority of the question. I was curious if any one knew of a form of wavelength frequency( which is what these all are) where I can get a concrete number for the output.
No. As far as I know, there is no other form of the wavelength/frequency equation, a.k.a. the relativistic Doppler shift equation. And if there is any other form, there is no way it will give you a concrete (finite) number, simply because the actual wavelength/frequency in the situation you're asking about is not a finite number.
If one were to calculate the speed of a projectile then the answer for the speed of light leaving headlights traveling at the speed of light is c or 1 which makes sense and when running the numbers for the car and virtually anything else in this question...
... huh?
...then yes it is an "impossible situation" but I can not think any reason why you can not figure out what the Doppler shift would be of light if you input the relativistic speed of it's source as light.
The reason is right there in the equation you've been told about - it blows up. It gives an undefined result. You have to divide by zero. That's why you can't figure out the Doppler shift of light coming from a source traveling at the speed of light.

Alternatively, if you want a more physical reason, it's that the situation can never happen. Headlights can never be traveling at the speed of light.
As I think about it, when you give c the v of c this does not mean you have to break the speed of light and violate theory. So can someone explain to me why you can't or if it is in any way mathematically possible to figure this.
Sorry, but I don't understand what you're saying here at all...
I asked the question broadly to try and open it to any other unique(means I did not see it) suggestions or p.o.v. on this type of situation
Sure, it's good to be open to new ideas in general, but asking for unique suggestions or points of view is not necessarily physics - they need to be backed up with evidence. And the physical community can present plenty of experimental and mathematical evidence that it's impossible to cross the speed of light.