Driving a car with headlights on at the speed of light: another question

[elegysix]

So I remember in basic physics the scenario:

If you're driving at the speed of light, and you turn on your headlights, can you see them?
I know the standard answer is yes; that it propagates at speed c no matter the reference frame.


So how about this:

Suppose I'm driving my car towards a mirror at a speed v (slightly less than c), which is far off at some distance d. Suppose my headlights give off wavelength λ.

how long does it take me to see the reflection off the mirror?

how long does it take for my friend to detect the light at the mirror? (given he somehow knew precisely when the headlights were turned on)


What frequency of light do I observe coming from my headlights, and what frequency do I observe returning from the mirror?

what frequency is measured at the mirror?

I'm assuming the reflection will be blue-shifted from the perspective of the car.
Does that mean if I pulse my headlights for a time T, that the reflection will be a shorter pulse of time T' ? (in order to conserve energy?)

 

[rorix_bw]

I thought this would be simple, but now I see SR/GR all over it. :-(

 

[elegysix]

I'm just curious you know... If I were riding in a spaceship at speed ~c, and I tried to answer this, I'd be like well - either the light will get there in half the time it takes us, or we'll slam into it together and pulverize my buddy trying to measure it. lol

 

[Rush2112]

The distance from u 2 the mirror gets lorentz contracted according to gamma, which is the lorentz factor l=√l-v^2/c^2.

D=r x t, so t which is time to see ur reflection is (d-gamma/300k)•2

 

[PeterDonis]

If you're driving at the speed of light, and you turn on your headlights, can you see them?

You can't drive your car at the speed of light to begin with, so the question as stated this way is not well-posed. See the Usenet Physics FAQ here:

http://www.desy.de/user/projects/Physics/Relativity/SpeedOfLight/headlights.html

I know the standard answer is yes;

No, the "standard" answer to the question as you stated it is what I said above.

that it propagates at speed c no matter the reference frame.

This part is OK, but there is no "reference frame" that moves at the speed of light.

Suppose I'm driving my car towards a mirror at a speed v (slightly less than c), which is far off at some distance d. Suppose my headlights give off wavelength λ.

Ok, this way of posing the question is well-defined, since v < c.

how long does it take me to see the reflection off the mirror?

According to your clock, or according to a clock that is at rest with respect to the mirror? We can do the calculation in either frame and then just apply the time dilation factor to get the answer in the other; let's try the latter frame first. In that frame, say you are a distance d from the mirror when you turn on the headlights. The beam will take a time d/c to travel to the mirror. When it reaches the mirror, you will be a distance d(1 - v/c) from the mirror (because you traveled a distance dv/c in time d/c). You are still moving to the right at v, and the reflected beam moves to the left at c, so it takes an additional time d(1 - v/c)/(v + c) for the reflected beam to return to you. That gives a total time d/c[1 + (c - v)/(c + v)], which simplifies to 2d/(v + c).

(A simpler way to get the same answer is to think of the beam as coming from your reflection in the mirror. That reflection is a distance 2d away from you, and the beam moves to the left from that point at c while you move to the right at v, so your "closure rate" is v + c.)

The above time is for a clock at rest with respect to the mirror; to get the time that elapses on your clock, just divide by the Lorentz gamma factor for v.

how long does it take for my friend to detect the light at the mirror?

Just to clarify, your friend is co-located with the mirror and at rest relative to it? If so, the time is d/c.

(given he somehow knew precisely when the headlights were turned on)

The only way he can know is to see the beam, unless you and he pre-arranged some scenario such that he could calculate in advance at what time, according to his clock, you would turn on the headlights.

What frequency of light do I observe coming from my headlights,

Assuming you had a measuring device just in front of your car, at rest relative to the car, it would measure the light coming from your headlights to have wavelength λ. (Since you specified wavelength above, it would be better to ask for answers in terms of wavelength; or else specify the headlight frequency instead.)

and what frequency do I observe returning from the mirror?

Since the mirror is moving towards you, you will see light returning from the mirror as blueshifted (higher frequency, shorter wavelength) compared to λ.

what frequency is measured at the mirror?

A measuring device at rest relative to the mirror will see you coming towards it, so it will measure the light from the headlights to be blueshifted by the same factor as you see the light returning from the mirror.

Does that mean if I pulse my headlights for a time T, that the reflection will be a shorter pulse of time T' ? (in order to conserve energy?)

As observed by you in the car, yes, the outgoing pulse will last for a longer time than the incoming reflected pulse.

 

[Antiphon]

If you slam into the mirror at just under the speed of light, is it 7 years of bad luck, 7*gamma or 7/gamma?

 

[elegysix]

What is the speed of my car as I would measure it from inside? If time is dilated for me, then observed objects would appear to move slower, right? (slower than the speed measured from a stationary frame)

suppose there were 'mile markers' I could see from the car - how long would I measure the time taken to pass between them? (say markers every c*60 m, as measured from a stationary frame)

If I then said v= d/t = c*60/t, then what speed would I think I'm traveling at? (just by observing the time(t) it takes me to pass between the markers.)

it would be less than the speed observed from a stationary frame, right?

 

[ghwellsjr]

No, you would measure the speed of the milestones passing you at the same speed that they measure you passing them. However, you will also measure the milestones to be closer together than one mile, so if you claimed that they were really one mile apart, you would conclude that you were traveling faster than you really were.

 

[elegysix]

So we're saying time is dilated, and the length is contracted. And we both measure the same speed.

if I use v=d/t

then d/t = d'/t' (' meaning moving frame)

if d > d' (contraction in ' frame)

then d/t = d'/t' iff t > t' , right?

 

[Dale]

So we're saying time is dilated, and the length is contracted. And we both measure the same speed.

Don't forget the relativity of simultaneity. The other guys time is dilated, length is contracted, and his clocks are desynchronized so that you both measure the same speed.

 

[jtbell]

What is the speed of my car as I would measure it from inside?

Zero.

 

[ghwellsjr]

So we're saying time is dilated, and the length is contracted. And we both measure the same speed.

if I use v=d/t

then d/t = d'/t' (' meaning moving frame)

if d > d' (contraction in ' frame)

then d/t = d'/t' iff t > t' , right?

What d and d' are you proposing in both frames? I presume in the milestone frame the distance between pairs of milestones is one mile and the milestone observer has synchronized clocks placed at each milestone and when you pass each one, a record is made of the time and then a calculation of your speed is made by taking the time difference between a pair of milestone markers and dividing that into one mile, correct?

But how do you measure the speed of the milestone markers as they pass you? I already said that if you use a clock traveling with you and you measure how long it takes to get from one milestone marker to the next and you assume that they are one mile apart, you will calculate that you are traveling faster than you really are. Do you understand why you cannot calculate your speed that way?

So what distance would you use and how would you measure the time? What if there was just one milestone marker, how would you measure its speed relative to you? Have you thought about this?

 

[elegysix]

lets make it simple - Say I'm very good at counting seconds in my head. so no clocks.

the distance, d, is the distance between markers relative to a stationary observer/frame.

I know the distance between markers, prior to getting in the car, as measured in the stationary frame. So whenever I pass a marker I start counting, and when I pass another I stop and punch in my calculator d/t.

What speed would I calculate? the same as a stationary observer?

 

[Doc Al]

lets make it simple - Say I'm very good at counting seconds in my head. so no clocks.

So your head will serve as a clock, just like any other.

the distance, d, is the distance between markers relative to a stationary observer/frame.

OK.

I know the distance between markers, prior to getting in the car, as measured in the stationary frame. So whenever I pass a marker I start counting, and when I pass another I stop and punch in my calculator d/t.

OK.

What speed would I calculate? the same as a stationary observer?

No. The quantity you calculate will equal gamma*v, where v is the car's velocity in the stationary frame. That quantity is sometimes called the 'proper velocity' (see: Proper velocity), but it's not the velocity of anything in the usual sense, since you're mixing quantities measured in different frames.

 

[elegysix]

No. The quantity you calculate will equal gamma*v, where v is the car's velocity in the stationary frame.

Ok. thanks for clearing it up.


Quick question though,

If there's an atomic clock on my car, I can see how the path of the light needs to be longer, like they show in the section called "Simple inference of time dilation due to relative velocity" at http://en.wikipedia.org/wiki/Time_dilation

But, they don't mention the length contraction. shouldn't it be used to correct D?

 

[ghwellsjr]

That would be covered under the article on Length Contraction:

http://en.wikipedia.org/wiki/Length_contraction

But it's not used to correct D--why do you ask that?

 

[elegysix]

we just talked about how the distances between markers would be contracted, so wouldn't the distance the mirrors move in the time deltaT' be contracted?

what v are they using? shouldn't it be multiplied by gamma or something?

 

[ghwellsjr]

Mirrors? I thought there was just one. Are you now thinking of each milestone as having a mirror?

Are you thinking that you have to correct the distance that you measure between milestones by gamma in order to get the correct velocity? If so, don't concern yourself with gamma. Just come up with a specific way to measure the speed of an object approaching you using your own measurement devices. Think about how it can actually be done in real life.

 

[elegysix]

I was asking about this diagram in reference to an atomic clock traveling at speeds near c.

they show the bottom length of the right triangle as 1/2vΔt'

since we are viewing it from a different frame, shouldn't that length be contracted?

 

[Doc Al]

I was asking about this diagram in reference to an atomic clock traveling at speeds near c.

You mean a 'light clock', not an atomic clock.

they show the bottom length of the right triangle as 1/2vΔt'

since we are viewing it from a different frame, shouldn't that length be contracted?

No. That's the horizontal distance traveled by the clock. It's not the length of anything. In the rest frame of the clock, that distance is zero. Length contraction is irrelevant.

 

[ghwellsjr]

I was asking about this diagram in reference to an atomic clock traveling at speeds near c.

they show the bottom length of the right triangle as 1/2vΔt'

since we are viewing it from a different frame, shouldn't that length be contracted?

Only the lengths of the mirrors are contracted but that is of no consequence in illustrating how a light clock works when the mirrors are placed orthogonal to the direction of motion. If you rotated the light clock by 90 degrees so the light went forward to hit the first mirror and then came back to hit the second mirror, then the distance between the mirrors would have to be shown contracted in order for the orientation of this light clock to tick at the same rate as the one you showed, but that form of the light clock is very difficult to illustrate in a static picture so it is not often done.

Have you thought about how you would measure the speed of an approaching milestone?

 

[elegysix]

i'm still trying to justify length contraction to myself... It just doesn't sit well with me.
I can understand how things might appear contracted, but I just can't make sense of how it is a real, measurable contraction, and not just an optical illusion.

Surely if a measuring tape lay along my path, it would appear contracted too. What difference does it make if I see it differently? I travel some length along it, in some amount of time. Then it does not matter what I think - I'd have gone x distance in t time.

Why wouldn't I regard the observed contraction as an optical abberration? If I were seeing by sonar in water, traveling near the speed of sound in it - wouldn't I predict a similar contraction if I thought that the speed was invariant, like c?

I'm not trying to say everyone's wrong, I'm just trying to understand why my questions are wrong - in hopes to better understand what's going on.
Thank you guys for being patient with me

 

[ghwellsjr]

I don't think your questions are wrong, it's just that I think what will convince you is to focus on how you would measure the speed of an approaching object. Then you can have that object travel past you and you can time how long it takes to pass you, that is, how long it takes from the leading edge getting to you until the trailing edge gets to you, then based on its speed and the time interval, you can determine how long it is. There will be no optical effects involved in this measurement and yet you will measure that the object has been contracted. And that object can do the same thing for you and determine that you are contracted. It all depends on you each measuring the speed of the other one to be the same. So that's why I'm asking you to focus right now on how you would measure the speed of an approaching object.

 

[elegysix]

I suppose using a laser on the front of a directly approaching object, and measuring the blue shift of the reflection could be used to determine its speed. And the other observer could use the same. (I couldn't think of another way that didn't rely on my vision)

To time the passing I think I'd try using a laser to feed a sensor, perpendicular to the motion of the object. So that as it passes, the laser will be blocked, and after it passes, the feed will resume.But I can't go out and do this experiment. So how do I know that I will measure a contracted length?

 

[elfmotat]

To time the passing I think I'd try using a laser to feed a sensor, perpendicular to the motion of the object. So that as it passes, the laser will be blocked, and after it passes, the feed will resume.

This is good. When the object first crosses the laser, call it event A. Call the event where the object has completely passed (i.e. the laser is unblocked) event B. Now try looking at events A and B from the object's rest frame.

 

[ghwellsjr]

I suppose using a laser on the front of a directly approaching object, and measuring the blue shift of the reflection could be used to determine its speed. And the other observer could use the same. (I couldn't think of another way that didn't rely on my vision)

I'm confused. If there is "a laser on the front of a directly approaching object", where would be the reflection?

To time the passing I think I'd try using a laser to feed a sensor, perpendicular to the motion of the object. So that as it passes, the laser will be blocked, and after it passes, the feed will resume.

This is a different laser, correct?

But I can't go out and do this experiment. So how do I know that I will measure a contracted length?

Let's take one step at a time, let's just focus on how you would measure the speed of an approaching object.

 

[elegysix]

I'm confused. If there is "a laser on the front of a directly approaching object", where would be the reflection?
Let's take one step at a time, let's just focus on how you would measure the speed of an approaching object.

To measure the speed of an approaching object:
take a laser of known λ, point it at the approaching object, and measure λ of the reflected beam.

Might as well say the approaching object has a mirror on the front of it, which reflects my laser back to me. The reflection will be blue shifted, and I can calculate the velocity from the change in λ.


And yes, there are two different lasers here. The one I just explained, and the other for measuring how long it takes to pass.

 

[ghwellsjr]

To measure the speed of an approaching object:
take a laser of known λ, point it at the approaching object, and measure λ of the reflected beam.

Might as well say the approaching object has a mirror on the front of it, which reflects my laser back to me. The reflection will be blue shifted, and I can calculate the velocity from the change in λ.

And yes, there are two different lasers here. The one I just explained, and the other for measuring how long it takes to pass.

Yes, that is a good way to measure the speed of an approaching object. But you also mentioned that the other observer could do the same thing, so if he has an identical laser pointing at your mirror, then couldn't you also just detect the blue shift of his laser to measure his speed? Now you have two different ways to measure his approaching speed and they should give the same answer, correct? Have you thought about the two different calculations you would make to come up with the same speed?

 

[elegysix]

both lasers would be shifted by the same amount. Giving the same speeds.

Seeing as length contraction is a result of time dilation, without assuming either of those, I can't see any difference in the calculations of the two speeds.

Is there something I've missed?

 

[ghwellsjr]

Yes, you missed the fact that the approaching observer will see your laser as blue shifted (higher frequency) and the reflection of that beam will be just like if he had another laser tuned to that higher frequency. So you will see the reflection of your laser with more blue shift than you will see in his laser beam, correct?