Why "light clock" is flawed.

gnomechompsky

I have been considering this thought experiment for some time, and the principle seems "broken". I'm wondering if by playing devil's advocate you can show me what is going wrong.

-In SR, the light clock was used to predict time dilation in a "moving" object.

-According to the Twin Paradox, two distinct frames of reference in different inertial motion would see time dilating in the other one, because all motion is relative.

-The counter argument to this is that the frames of reference are not identical, because in the Twin Paradox, the Twin flying away from earth accelerates and is not in constant inertial motion.

-We therefore can't use SR to answer the Twin Paradox, we have to use GR.

-Using a GR thought experiment, the Light Clock would no longer function if acceleration is great enough. The acceleration would cause the beam of light to bend outside the light clock, even from the light clock's frame of reference.

HallsofIvy

You can use SR even when acceleration is involved by doing a different "SR" calculation for each instant, treating the speed at that instant as constant.

Mentz114

Special relativity is the theory of Minkowski spacetime, which includes accelerating observers, see Rindler chart,en.wikipedia.org/wiki/Rindler_coordinates and also this is very interesting http://gregegan.customer.netspace.ne...erHorizon.html

AJ Bentley

Simply not true.

Acceleration plays no part in the resolution of the paradox.
In fact, the whole point of the paradox from a tuition point of view is to get student's to realise that there is no paradox.

The point is to illustrate the absurdity of our ordinary concept of simultaneity - which is what the 'paradox' relies on for it's puzzle effect.

yuiop

As others have mentioned SR can handle acceleration of the observers and so it not a GR thought experiment, but it is related to GR via the equivalence principle. It is true that the light beam will bend during acceleration and miss the mirror or the receiver on the way back in a traditional transverse light clock which is not ideal for the accelerating case. If you wish to analyse time dilation using a light clock in the accelerating case it would be best to have a light clock that is orientated parallel to the acceleration. One obvious limitation of such a clock is that when the signal leaves the source the velocity of the source is v, and when the signal returns the velocity is v+a*t, so the light clock is not measuring the time dilation corresponding to velocity v except in the case the the light clock and the a*t term is vanishingly small, so the accuracy of the clock is size dependent. We could design the clock so that it has one source, but two mirrors in opposite directions so that when the clock is at rest the signals return simultaneously. When the light clock is accelerating parallel to its length, I don't think the signals will return simultaneously (but I have not done the calculations yet) and so you could obtain some sort of average time from the two oppositely orientated clocks.

gnomechompsky

Please explain this because every resolution of the Twin Paradox I have seen refers to acceleration. If both frames of reference are otherwise in relative inertial motion what is the difference between them qualitatively?

Mentz114

The twin 'paradox' requires the twins to start in the same place and meet up again later. The elapsed time on their clocks is equal to the proper-time of their worldlines. The proper time can be calculated simply and has nothing to do with 'time dilation' effects observed during the trips. The proper time takes into account the 4-D path and so any changes in velocity will be reflected in the proper time. See http://en.wikipedia.org/wiki/Proper_time

Rasalhague

There must be some point on the path through spacetime of the twin who's younger when they meet where this twin's velocity changes. In practice, this would entail acceleration. I think what AJ Bentley means is that you can get some intuition about the scenario by considering a simplified, not physically realistic, case where, instead of the path having a curved segment, representing acceleration, you consider the outward journey and the journey home as two different straight line segments, and you treat the change in velocity where the twin reverses direction in space as an abrupt corner connecting these line segments. So in this simplified model, there's not really acceleration in the sense of a derivative of velocity, because the velocity isn't continuous, and therefore not differentiable, at the corner, although you might hear some people refer to it informally as an "infinite acceleration".

Bussani

I got some good replies when I asked a question about this myself. It might be helpful to you.

http://www.physicsforums.com/showthread.php?p=2549150

yuiop

Let us say for the sake of argument that acceleration does affect time dilation and see how that affects the twins paradox resolution.

Consider one twin accelerating from 0 to 0.8c in one second and then traveling at a constant velocity for 10 years as measured in the Earth frame) before turning around in the space of 2 seconds and returning at 0.8c and finally coming to a stop by decelerating to zero in 1 second and re-uniting with his twin. The total journey time is 20 years (+4 seconds) and the SR time dilation factor at 0.8c is 0.6 so the travelling twin has aged roughly 12 years in the time the stay at home twin has aged 20 years. Now the acceleration takes place in a total of 4 seconds. If time stops still during the acceleration (worst case) then the largest error that is introduced by ignoring the time dilation during the acceleration phase in this example is 4 seconds and that fails to account for why the travelling twin has lost a total of 8 years compared to his stay at home twin.


DrGreg also produced a very nice diagram that makes it very intuitively clear why acceleration is not the explanation in the twins paradox here: http://www.physicsforums.com/showpos...55&postcount=4

Lastly, as I mentioned before, analysis of a light clock parallel to the axis of motion would give us an informative insight into how an ideal clock performs during acceleration. This should not be too difficult to calculate for constant acceleration. Are you interested in that aspect or just acceleration as it applies to the twins paradox?

gnomechompsky

-This doesn't solve the twin paradox. If we pretend acceleration plays a negligible impact, using the light clock thought experiment then both observers will see the same amount of time dilation in the other observer.

-You are stating that time dilation over the course of the 10 years is caused by this inertial motion, but the formula from this is derived from the light clock thought experiment which we have resolved on the basis of the twin paradox yet.

Passionflower

Acceleration does effect the rate a clock ticks with respect to one that does not accelerate. So the longer the time interval the greater the time differential.

yuiop

That may be true, but I have shown in the last post, that the time dilation due to acceleration in the twins paradox can be reduced to a negligible error, e.g less than 4 seconds "lost" due to acceleration time dilation compared to 8 years "lost" due to constant velocity.

It can also be shown that if an object is accelerating away from us, its instantaneous clock rate relative to our clock, is a function of its instantaneous velocity relative to us (i.e t' = t*sqrt{1-v^2/c^2) where v is its instantaneous velocity, and is independent of its acceleration.

yuiop

Yes, I am assuming the time dilation equations of SR are correct. What is important in the twins paradox is the path length through spacetime. On the outward trip of one of the the twins, there is some ambiguity of the time dilation of the twins relative to each other. When the paths are plotted on a spacetime diagram, it is true that the path of the Earth twin looks longer from the travelling twins frame and vice versa. There is always some ambiguity in clock rates when the clocks are not at rest with one another. However, after the travelling twin returns home, and the two clocks are alongside each other again, there is no ambiguity. The path of the travelling twin through spacetime is ALWAYS longer than the Earth twins path, from the point of view of ANY inertial observer. If we define relative time dilation in terms of spacetime path lengths, then the ambiguities can be made to disappear when the two clocks come to rest wrt to each other even if they are spatially separated.

We can do the twins paradox in anther way that removes all acceleration from the problem. Consider one observer A that remains at location A (EDIT: A remains at rest in frame A). Another observer B is passing A and moving at constant velocity v relative to A. A and B synchronise clocks so that they read the same time. B travels for some distance away from A, until he passes another observer C moving with velocity -v towards A. C adjusts here clock so that it reads the same as B's clock at the instant they pass each other. All observers agree that at the passing event, when C synchronises her clock with B, that C's clock reading is a good representation of the elapsed time on B's clock. When C passes A, C and A compare elapsed times at the passing event and agree that the elapsed time on C's clock is less than the elapsed time on A's clock. No one has accelerated at all during this thought experiment, so the acceleration aspect has been completely removed.

Mentz114

Time dilation has nothing to do with what the clocks will read when they meet !!!
There's no way to understand this generally by looking at bits of journeys and talking about 'changing frame'. The fact is that what is shown on a clock is the proper time of the worldline. The proper time takes into account the details of the trip in 4 dimensions and always gives the correct answer. It's got nothing to do with whether one traveller is inertial or not, although it's perhaps a little simpler. Talking about acceleration doesn't help much eaither. The proper time takes into account all changes of velocity on the worldline.

yuiop

I thought I would elaborate a little on this unsupported statement.

First lets take the equation for the instantaneous velocity on a accelerating rocket with constant proper acceleration (a) at time (t) in the non accelerating frame:

v = at / sqrt[1 + (at/c)2]

as given by Baez http://www.xs4all.nl/~johanw/PhysFAQ...SR/rocket.html

We can solve this equation for (at) to obtain:

at = v/sqrt[1-(v/c)^2]

Baez also gives the instantaneous gamma factor of the accelerating rocket at time t as:

γ = sqrt[1 + (at/c)^2]

and if we substitute the symbolic value for (at) obtained earlier into the equation for gamma immediately above we get:

γ = sqrt[1+((v/c)/sqrt[1-(v/c)^2]] = 1/sqrt[1-(v/c)^2]

demonstrating that the instantaneous time dilation of an accelerating particle can be formulated in such a way that it is only a function of the instantaneous velocity at any given instant and independent of the acceleration.

Passionflower

I do not see any problem with what you write Kev it is as you say: it all depends on the relative speed. But what causes a relative speed to change? Right, inertial and proper acceleration. So what drives a change in clock rates?