Time Dilation: When v = c

[adjective]

Hi all,

 

I've been working my way through the math of special relativity to get a grasp on it and such.

 

Anyway, when I came across the time dilation equation (t' = t / sqrt(1 - v^2/c^2)) I wondered, given the nature of the subject, what happened to time when the velocity was the speed of light. Basically, I've come across two "interpretations." The first simply says that time stops, as observed by an outside frame. The other sort of glosses over the subject and doesn't mention what happens, instead leaving it to the reader to interpret (often they lead this interpretation by listing time dilation factors as v approaches c, leading the reader to think that time would cease).

 

The second approach is simply misleading in that it leads the reader into concluding something without actually stating so. The first bothers me on mathematical grounds.

 

If v = c, then v^2/c^2 = 1. If so, then the denominator is sqrt(1 - 1) = sqrt(0) = 0. So, the answer is undefined.

 

I tried taking the limit as v approached c, but I couldn't remove the discontinuity.

 

If anyone can shed any light on this--like if my algebraic skills are too weak to remove the discontinuity, or if there's a generally accepted interpretation on the subject--I would be extremely appreciative.

 

Thanks.

 

(Edit: Equation typo.)

[infamous]

Hi all,

 

I've been working my way through the math of special relativity to get a grasp on it and such.

 

Anyway, when I came across the time dilation equation (t' = t / sqrt(1 - v^2/c^2)) I wondered, given the nature of the subject, what happened to time when the velocity was the speed of light.

 

If anyone can shed any light on this--like if my algebraic skills are too weak to remove the discontinuity, or if there's a generally accepted interpretation on the subject--I would be extremely appreciative.

 

Thanks.

 

(Edit: Equation typo.)

 

Look at it this way adjective, t' is equal to the passage of time on earth. And t is equal to the passage of time on a hypothetical rocket which travels at a significant percentage that of c. As v approaches the speed of light the passage of time on the rocket will slow with respect to those left on the earth. For those on the rocket, time will appear to pass at a normal rate but compared with those left on earth they will age much more slowly. No need to ever worry about v reaching that of c, for this is a physical impossibility. But just for the sake of argument, if the rocket could travel at the speed of light, those on board would still experience the passage of time relative to their own existence but they could never return to earth because an eternity of time would have passed there. The fact remains that not even the smallest amount of mass will ever travel the speed of light.

[UncleAl]

First, lightspeed (the point of view of a photon or other rest mass=0 particle) is not an inertial reference frame. Second, no massed body may travel at lightspeed. There is no paradox because your situation cannot arise. Consider two observers rushing head-on at each other. In each observer's reference frame he is traveling at 0.9999 lightpseed vs. the fixed stars. How fast are they traveling vs. each other?

 

Given any achievable velocities V1 and V2 and any finite lightspeed, the bound on the relative velocities of V1 and V2 as viewed by any inertial observer cannot exceed

 

(V1 + V2)/[1 +(V1)(V2)/c^2]

 

This is transformation of velocities parallel to the direction of motion. For velocities at an arbitrary angle theta, Jackson gives

 

u_parallel = (u'_parallel + v)/(1+(v dot u')/c^2)

u_perp = u'_perp/(gamma_v(1+(v dot u')/c^2))

 

http://www.physics.umanitoba.ca/~souther/waves02/feb0402/sld011.htm

 

SR is a self-consistent geometry. No matter how you swing it, no paradox obtains.

[Qfwfq]

Uncle Al, where did Adjective talk about a body with mas travellng at c? Where di he talk about a paradox?

[Qfwfq]

If v = c, then v^2/c^2 = 1. If so, then the denominator is sqrt(1 - 1) = sqrt(0) = 0. So, the answer is undefined.

 

I tried taking the limit as v approached c, but I couldn't remove the discontinuity.

The discontinuity is not eliminatable, the limit is infinity. This means that, the more v approaches c, the greater the ratio between the time coordinates.

[Boerseun]

Pardon my ignorance (UncleAl - don't burn me, pleeze :wave: ), but is it safe to say that a photon exists out of time? Seeing as it's massless, v=c with no funnies happening, and v can hypothetically exceed c seeing as there's no asymptotic approach to c with mass increasing.

 

Yes, I know - c is a constant, blah di blah, just a thought...

[Tormod]

is it safe to say that a photon exists out of time?

 

*boom*

[adjective]

Thanks Boerseun, that's the exact physical analog I had in mind.

 

UncleAl: I'm not claiming any paradox. I realize there's a subtlety I must be missing and I'm wondering if anyone else can help me understand what it is.

 

In regards to your statement that the speed of light can't be an inertial frame of reference--the math is similar to the problem posed, right? Because you use Lorentz transformations and, thus, get an undefined answer.

 

If that's the case, are people just plain wrong in saying, "Time would stop at light speed to an outside observer?" Would they be better off just saying that it was undefined and, as such, completely unknown what happens to time for photons?

[Qfwfq]

If that's the case, are people just plain wrong in saying, "Time would stop at light speed to an outside observer?" Would they be better off just saying that it was undefined and, as such, completely unknown what happens to time for photons?

Well, not quite wrong. It makes sense to say that the photon's elapsed proper time is zero although, in our coordinates, it travelled here from another galaxy in a few million years.

 

This has been verified in the v --> c limit with particles that decay, with a measured average decay time, such as muons.

[infamous]

Well, not quite wrong. It makes sense to say that the photon's elapsed proper time is zero although, in our coordinates, it travelled here from another galaxy in a few million years.

.

 

Interesting thought Qfwfq, the photon's existence being only an instant old while the material universe ages an eternity.

[quantum quack]

is it worth noting that when v=c the length of the dimension is zero, therefore time is non-existant.

Possibly this length contraction helps remove the discontinuuity so time and distance = 0

please excuse me if I am way of the mark.

 

edit:

possibly also it is worth noting that time is siupposed to be infinite duration at 'c' and not zero duration, although the actual observation would be zero distance but infinite time to travel a distance of zero. Thus you have your so called "paradox'

[Qfwfq]

possibly also it is worth noting that time is siupposed to be infinite duration at 'c' and not zero duration, although the actual observation would be zero distance but infinite time to travel a distance of zero. Thus you have your so called "paradox'

:eek:

 

The elapsed proper time is zero, whereas a finite proper time such as the average decay time would be infinite in the lab coordinates. This means that we couldn't see a massless particle as having a finite decay time. This consideration comes into the speculations about the neutrino being massless or having a tiny mass.

 

No paradox.

[quantum quack]

so Q yo are basically saying that one has to consider the reference frame perspective properly and thus no paradox would be evident?

 

say from an imagnary masssless particle with v at c according to another observer would have according to that particles observer a v=0

or in other words,

to us it has v=c but to it it has v=0

thus it is at rest according to it's RF which leads us back to the SRT position on rest frames.

edit:

Oops I think I just confused the sh*t out of every one...sorry

[EWright]

My it is difficult to get a straight answer sometimes. I've had the same experience with a few of my posts. To answer your question directly, special relativity demands that a photon experiences NO time; or that time STOPS when you reach light speed. The reason you can't get a definitive answer is because they're getting too caught up in the "fact" that no object of mass can reach c. Because that's impossible doesn't mean it's impossible to answer the question hypothetically. And for a photon, in which case the question is not hypothetical, the answer based on SR is that no time passes for a photon and that thus, light does not age. Now I will also tell you that SR is wrong about this, but that is what it predicts.

[quantum quack]

another question that may be worth askign that is relevant is :

"At what velocity does an object of mass become massless?"

 

Impossible to answer I would think.

[EWright]

another question that may be worth askign that is relevant is :

"At what velocity does an object of mass become massless?"

 

Impossible to answer I would think.

 

Does relativity not state that an object INCREASES in mass as it is accelerated? :)

[adjective]

EWright: What justification do you have behind the claim that SR demands an infinite time dilation at v = c?

 

quantum quark: As far as I know, mass is intrinsic to matter. No velocity will separate the two. I suppose there may be something in there with antigravity, but I do not know enough about that to make any assertions.

 

By definition, a massless particle travels at the speed of light. So, I suppose the question could be answered (albeit rather illogically) that an object becomes separated from its mass at the speed of light.