# Time Equation and Special Relativity

## Summary:

Time defined
If time slows as an object increases velocity wouldn't that indicate that time is object specific? And if the speed of light is a constant and does not change regardless of the velocity of an object wouldn't that indicate that time used to measure the speed of light changes? The video I have posted at the following link provides a graphical representation of the concept I am trying to convey:

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Orodruin
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You are failing to account for the relativity of simultaneity and basing all of your argumentation on the rest frame of E. This is a common misconception. Note that you could just as well have the rocket travelling in the opposite direction and the time dilation of the rocket would be the same. Relativity of simultaneity is crucial for the discussion of time dilation.

I suggest picking up a good introductory textbook on special relativity.

bhobba
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Summary:: Time defined

If time slows as an object increases velocity wouldn't that indicate that time is object specific?
The main problem with your analysis is that there is no such thing as absolute velocity. In other words, an object does not have a definite velocity; only velocity relative to something else.

Dale
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Summary:: Time defined

wouldn't that indicate that time is object specific?
Yes, this is called “proper time”, where proper is used to indicate ownership rather than correctness. As others have indicated, your formula is incorrect. The time dilation formula is a comparison between one reference frame’s coordinate time and an object’s proper time.

bhobba and russ_watters
Mister T
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Summary:: Time defined

If time slows as an object increases velocity
That oft-mentioned phrase is the source of a lot of confusion. It's an over-simplification because it seems to imply that all you need is one clock at rest and one clock in motion. But you need a third clock!

If I'm moving and you're at rest, then we can compare our clocks when I pass you. But that's it because we will never share the same location to ever be able to make another comparison and conclude that my clock is running slower than yours. What you need to do is have another clock located some distance away and at rest relative to you. You synchronize your two clocks, and as as I pass by your second clock we can make another comparison and then we can indeed conclude that, according to you, my clock is running slow.

Stick with me here. This is more than a pedantic digression.

I will not agree that your two clocks are synchronized, and will claim that that is the reason why you conclude that my clocks are running slow.

If I were to set up two clocks, both at rest relative to me, and repeat the above, I will conclude, for the same reasons that you did, that your clocks are running slow. And you will claim that I've reached that conclusion because I didn't synchronize my clocks correctly.

It is in this way that each of us will conclude that the other's clock is running slow.

Your video tells only half the story, and even that half is incomplete.

dsaun777, bhobba, SiennaTheGr8 and 2 others
Time doesn't slow as velocity increases, unless you are in a gravitational field, which has to do with acceleration rather than velocity. The behavior of time depends on the portion of velocity toward (or away from) an observer, and it can either run slower, or faster, depending on whether that motion is away from or toward the observer respectively.

If you are traveling away from me, I will observe your clock running slower than mine. If you are traveling toward me, however, I will observe your clock running faster than mine.

In common-sense (non-relativistic) terms, the observed time is doppler shifted by your speed toward or away from me, and this change is created and maintained by the travel time of light.

Acceleration, however, behaves differently. Your clock does run slower than mine while you are accelerating, and this includes the acceleration of being in a gravity well, regardless of whether you are actually moving relative to the source of the gravity. Motion relative to a gravity well increases the acceleration, and it increases it in proportion to the strength of the field.

Vanadium 50, weirdoguy, Dale and 1 other person
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Time doesn't slow as velocity increases, unless you are in a gravitational field, which has to do with acceleration rather than velocity. The behavior of time depends on the portion of velocity toward (or away from) an observer, and it can either run slower, or faster, depending on whether that motion is away from or toward the observer respectively.

If you are traveling away from me, I will observe your clock running slower than mine. If you are traveling toward me, however, I will observe your clock running faster than mine.

In common-sense (non-relativistic) terms, the observed time is doppler shifted by your speed toward or away from me, and this change is created and maintained by the travel time of light.

Acceleration, however, behaves differently. Your clock does run slower than mine while you are accelerating, and this includes the acceleration of being in a gravity well, regardless of whether you are actually moving relative to the source of the gravity. Motion relative to a gravity well increases the acceleration, and it increases it in proportion to the strength of the field.
This is all wrong.

Relativity, time dilation and differential ageing are not based on the delay in light signals reaching an observer. Velocity-based time dilation depends only on the relative speed between two observers or reference frames. It's independent of the direction of the relative motion and is symmetric, in the sense that both observers measure the same time dilation for clocks in the other reference frame.

Acceleration does not cause time dilation. This has been discussed many times on the forum.

Gravitational time dilation is dependent on differences in gravitational potential; not on the strength of the gravitational field.

If the consensus here is that acceleration doesn't cause time dilation, then the consensus here is wrong on a pretty fundamental and basic part of relativity. The Equivalence Principle has something to say about the relativistic difference between gravity and acceleration, namely that there isn't one.

And the equation doesn't stop working based on the direction of motion, but the factor has a value greater than 1 when the direction of motion is towards, and less than 1 when the direction is away. Which is to say, if you observe a clock moving away from you, it will tick more slowly, and if you observe a clock moving toward you, it will tick more quickly. And if you synchronize time with the clock, and then you separate, and meet again, the final difference in time will be entirely dependent on acceleration experienced in the intervening time, not velocity.

You can play with the integral to verify this yourself, if you want. You'll find it impossible to construct a path through space-time in which the final time dilation factor between two objects which synchronize and meet again isn't precisely proportional to the acceleration experienced on their paths.

PeroK
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If the consensus here is that acceleration doesn't cause time dilation, then the consensus here is wrong on a pretty fundamental and basic part of relativity.
You think that's likely then? That a forum moderated by experienced physicists lacks a basic grasp of relativity? Or, is it more likely that you've learned a bit of relativity and got some of the basics wrong?

Pencilvester
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You can play with the integral to verify this yourself, if you want. You'll find it impossible to construct a path through space-time in which the final time dilation factor between two objects which synchronize and meet again isn't precisely proportional to the acceleration experienced on their paths.
There's a discussion of the role of acceleration in the twin paradox here:

In effect, you can run the twin paradox without any acceleration, by recording clock readings for the outward and return journeys, using two physical clocks rather than one. In this way the out and back path through spacetime involves no physical acceleration.

The important thing for the original poster to know is that clocks can go more quickly, as well as more slowly, depending on whether travel is toward or away from.

This particular argument is largely irrelevant to that point.

Dale and weirdoguy
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The important thing for the original poster to know is that clocks can go more quickly, as well as more slowly, depending on whether travel is toward or away from.

This particular argument is largely irrelevant to that point.
This is irrelevant to relativity. Time dilation is the same regerdless of direction of motion.

The delay in light signals is an experimental detail, accounted for long before Einstein came along. In fact, the first good estimate of the speed of light came from observing that the period in the orbit of a moon of Jupiter was out of synchronisation depending on how far Jupiter was from the Earth. See here, for example:

https://en.wikipedia.org/wiki/Speed_of_light#First_measurement_attempts

This delay is not what special relativity deals with. Time dilation is a conseqence of a theory of spacetime and is not based on the experimental delay in light signals reaching an observer.

Janus
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The important thing for the original poster to know is that clocks can go more quickly, as well as more slowly, depending on whether travel is toward or away from.

This particular argument is largely irrelevant to that point.
If you are watching a clock approaching it will appear to run fast. However, that is due to the fact that the distance between it and you is decreasing and so is the light travel time. If you factor this effect out, you end up with determining that the approaching clock is running slow compared to your own. Time dilation is what is left over after you account for light travel time between the clock and yourself.

vanhees71, russ_watters and PeroK
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You can play with the integral to verify this yourself, if you want. You'll find it impossible to construct a path through space-time in which the final time dilation factor between two objects which synchronize and meet again isn't precisely proportional to the acceleration experienced on their paths.
Trivial counter example:

A pair of twins start on Earth at rest with respect to one another. Twin 1 accelerates to 0.6c, travels inertially for 1 year by his clock, reverses course and speed, travels inertially for 1 year (again by his clock, so returning to Earth) where he waits.

Twin 2 does the exact same thing, but travels for 2 years each way.

The accelerations experienced by the twins are identical, but their ages are different - twin 1 is 4.5 years older while twin 2 is only 4 years older (edit: at the time twin 2 returns, that is - so twin 1 is six months older than 2 thereafter, assuming no more relativistic travel). This is because acceleration has nothing to do with this, except inasmuch as it is needed to change velocity.

Note that I've implicitly used instantaneous acceleration. The same is true with a finite acceleration, but the maths is messier.

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Dale and PeroK
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If the consensus here is that acceleration doesn't cause time dilation, then the consensus here is wrong on a pretty fundamental and basic part of relativity. The Equivalence Principle has something to say about the relativistic difference between gravity and acceleration, namely that there isn't one.
Yes there is an equivalence between acceleration and gravity. But how this effects clocks isn't what you seem to think it is. It doesn't have anything to do with the difference in local gravity, but the difference in gravitational potential. So for instance, if you had a uniform gravity field ( one that does not change in strength with height), and placed two clocks at different heights in it, the higher clock will run faster, even though the clocks experience exactly the same gravitational acceleration.

Now while we don't have uniform gravitational fields to test this, we have another means to show this effect.
The surface gravity on Uranus is just a tad weaker than that on the Earth. However, Relativity predicts that a clock on Uranus would tick slower than one on the Earth rather than the other way around.

If you accelerate a clock in free space, your determination of its tick rate will only depend on its speed relative to you at any given moment and will not in any way depend on the value of its acceleration. This has actually been tested by using high speed centrifuges to produce extremely high g forces. The result is that any time dilation shown was only due to the speed of the sample and was not effected by the amount of acceleration it experienced.
If you have two accelerating clocks, accelerating in such a manner that, according to the clocks, one clock maintains a constant distance ahead of the other, the lead clock will run faster according to both clocks. This is equivalent to the uniform gravity example above and where the equivalence principle plays a role.

Dale, Ibix and PeroK
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If the consensus here is that acceleration doesn't cause time dilation, then the consensus here is wrong on a pretty fundamental and basic part of relativity. The Equivalence Principle has something to say about the relativistic difference between gravity and acceleration, namely that there isn't one.
You are reaching the wrong conclusion with your argument. First, the clock hypothesis in SR states that there is no additional time dilation depending on acceleration, only based on velocity. This has been validated experimentally at up to about 10^18 g. Second, applying the Equivalence Principle allows us to go from SR to local gravitational effects. So the absence of time dilation due to acceleration directly implies the absence of time dilation due to gravitational acceleration. In fact, gravitational time dilation is due to gravitational potential, not gravitational acceleration. So your direction of applying the Equivalence Principle was backwards and led you to the wrong result for both gravitational and kinematic time dilation.

And the equation doesn't stop working based on the direction of motion, but the factor has a value greater than 1 when the direction of motion is towards, and less than 1 when the direction is away. Which is to say, if you observe a clock moving away from you, it will tick more slowly, and if you observe a clock moving toward you, it will tick more quickly.
The important thing for the original poster to know is that clocks can go more quickly, as well as more slowly, depending on whether travel is toward or away from.
This is wrong. You appear to be confusing time dilation and the Doppler effect. Time dilation is different from the Doppler effect. To emphasize that, it is sometimes called the transverse Doppler effect. Time dilation is independent of direction. This is obvious from the formula:
$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$The velocity term is ##v^2=\vec v\cdot \vec v=|\vec v|^2##, so it depends only on the magnitude and does not depend on the direction.

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Mister T
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The important thing for the original poster to know is that clocks can go more quickly, as well as more slowly, depending on whether travel is toward or away from.
Look at the equation for time dilation in special relativity: ##\Delta \tau=\Delta t\ \sqrt{\frac{1}{1-v^2/c^2}}##, where ##\Delta \tau## is the proper time that elapses between two events and ##\Delta t## is the dilated time that elapses between those same two events. Note that ##\Delta \tau## is always smaller than ##\Delta t## except in the trivial case where ##v## is zero. All that matters is the speed ##v##, whether the direction of the velocity is towards or away has no effect.

phinds