# Generalization of SR

#### Päällikkö

Homework Helper
How has it been shown that time dilation is not a property of light? I've only seen the equation derived from a light clock or something dealing with light. How can time dilation (along with other predictions of special relativity) be generalized to everything as a property of time itself?

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#### Symbreak

Time is about measurement. If you measure a process to go more slowly, you can infer 'time' to slow down. In special relativity, it is for inertial observers, that move close to the speed of light, when processes start to slow down. The Lorentzian time dilation only applies for velocities near the speed of light - hence the necessity of refering to a photon. There is no dilation or legnth contraction associated with a poton though: only to the observer(s), relative to one another.
We can generalise special relativity by condisering accelerating frames: a light ray must appear to 'bend' between a reference frame that accelerates and one which is at rest... but really this is a conequence of it following a straight line in a non-euclidean geometric framework.

#### pervect

Staff Emeritus
Science Advisor
Päällikkö said:
How has it been shown that time dilation is not a property of light? I've only seen the equation derived from a light clock or something dealing with light. How can time dilation (along with other predictions of special relativity) be generalized to everything as a property of time itself?
Muons generated in the upper atmoshphere by cosmic rays have such a short lifetime that they could not reach the Earth's surface even if they moved at the speed of light before they decayed.

However, muons are observed at the Earth's surface.

The explanation for this is that they don't decay as quickly because of time dilation.

Other particles routinely generated by high energy particle accelerators are also moving at relativistic velocities, and show the same effect of time dilation on their lifetimes.

#### Aer

pervect said:
Muons generated in the upper atmoshphere by cosmic rays have such a short lifetime that they could not reach the Earth's surface even if they moved at the speed of light before they decayed.

However, muons are observed at the Earth's surface.

The explanation for this is that they don't decay as quickly because of time dilation.

Other particles routinely generated by high energy particle accelerators are also moving at relativistic velocities, and show the same effect of time dilation on their lifetimes.
I am familiar with this experiment, however I do have a question concerning the time dilation as you interpret it. First of all, the muon is moving constantly until it decays and never decelerates, correct? Since this muon never decelerates to the Earth frame in which it is observed, how can time dilation be observed by the muon if we are to assume that either the muon or the Earth can be considered at rest? That is, to the muon, the Earth's time is dilated - but to the Earth, the muon's time is dilated by the fact that it is shown to live longer than a muon at rest in the Earth frame would. If you go to my thread on special relativity, this is exactly opposite the view touted by posters there in which the moving frame doesn't agree that it's time is dilating until it accelerates to another frame that says that the previously moving frame was dilated.

#### JesseM

Science Advisor
Aer said:
I am familiar with this experiment, however I do have a question concerning the time dilation as you interpret it. First of all, the muon is moving constantly until it decays and never decelerates, correct? Since this muon never decelerates to the Earth frame in which it is observed, how can time dilation be observed by the muon if we are to assume that either the muon or the Earth can be considered at rest? That is, to the muon, the Earth's time is dilated - but to the Earth, the muon's time is dilated by the fact that it is shown to live longer than a muon at rest in the Earth frame would. If you go to my thread on special relativity, this is exactly opposite the view touted by posters there in which the moving frame doesn't agree that it's time is dilating until it accelerates to another frame that says that the previously moving frame was dilated.
In the muon's frame the earth's clocks are running slow, but the distance from the upper atmosphere to the surface is also shrunk due to Lorentz contraction, so in this frame you should still predict that the muon will reach the surface before its onboard "decay clock" runs out. The whole point of relativity is that you should get the same answer to all physical questions no matter which inertial frame you use, and the question of what a clock will read at the moment it reaches the same position as another physical object (like the surface of the earth) is such a physical question.

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#### Perspicacious

Päällikkö said:
How has it been shown that time dilation is not a property of light? I've only seen the equation derived from a light clock or something dealing with light.
Special relativity is derived without reference to light in the following papers:

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000043000005000434000001 [Broken]
http://arxiv.org/PS_cache/physics/pdf/0302/0302045.pdf [Broken]
Reference 3

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#### Aer

JesseM said:
In the muon's frame the earth's clocks are running slow, but the distance from the upper atmosphere to the surface is also shrunk due to Lorentz contraction,
According to a muon moving through the atmosphere as previously described (I'm not sure of the exact speed, I suppose I could look it up...) what is the length of the known universe? Let's assume the known universe is 15 billion light years across.

#### JesseM

Science Advisor
Aer said:
According to a muon moving through the atmosphere as previously described (I'm not sure of the exact speed, I suppose I could look it up...) what is the length of the known universe? Let's assume the known universe is 15 billion light years across.
The size of the observable universe is actually a lot larger than 15 billion light years due to the expansion of space--about 156 billion light years according to the latest estimates. http://www.glenbrook.k12.il.us/gbssci/phys/Class/relativity/u7l2e2muon.html [Broken] says a muon created in the upper atmosphere travels at around 0.998c relative to the earth, which gives a length contraction factor of around 0.0632. So, if the distance between the furthest points that we can see along the muon's axis of motion is 156 billion light years in our frame, in the muon's frame the distance between these points would be 9.86 billion light years according to a straightforward SR calculation, although this probably isn't completely legitimate when dealing with GR where spacetime can't be treated as flat as is assumed in SR (and even if space is pretty close to flat, the expansion of space means that spacetime is curved).

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#### Aer

JesseM said:
The size of the observable universe is actually a lot larger than 15 billion light years due to the expansion of space--about 156 billion light years according to the latest estimates. http://www.glenbrook.k12.il.us/gbssci/phys/Class/relativity/u7l2e2muon.html [Broken] says a muon created in the upper atmosphere travels at around 0.998c relative to the earth, which gives a length contraction factor of around 0.0632. So, if the distance between the furthest points that we can see along the muon's axis of motion is 156 billion light years in our frame, in the muon's frame the distance between these points would be 9.86 billion light years according to a straightforward SR calculation, although this probably isn't completely legitimate when dealing with GR where spacetime can't be treated as flat as is assumed in SR (and even if space is pretty close to flat, the expansion of space means that spacetime is curved).
Yes I agree with the comment about GR. However, the same situation can be applied in a region of space where GR is negligible and the same results should apply. Does this mean that the muon can see points further in the universe than the Earth?

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#### JesseM

Science Advisor
Aer said:
Yes I agree with the comment about GR. However, the same situation can be applied in a region of space where GR is negligible and the same results should apply. Does this mean that the muon can see points further in the universe than the Earth?
I'm not too well-versed in how cosmological horizons work...does anyone know if two observers who are currently at the same location in space but moving apart with a large velocity will have the same horizon or not?

#### Päällikkö

Homework Helper
Perspicacious said:
Special relativity is derived without reference to light in the following papers:

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000043000005000434000001 [Broken]
http://arxiv.org/PS_cache/physics/pdf/0302/0302045.pdf [Broken]
Reference 3
Thank you, this is exactly what I was looking for.

Another question (actually, several questions, but they're pretty much about the same thing) about relativity (this time GR, which I know little about):
Does light gain energy as it approaches a massive object?
What happens to light trying to "escape" from a black hole? Does it lose all its energy?
Is the light leaving the sun seen with a lower frequency from earth (does light lose energy as it leaves a massive object)?

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#### JesseM

Science Advisor
Päällikkö said:
Thank you, this is exactly what I was looking for.
Note that if you try to derive relativity without referring to light, you'll get an equation that, depending whether a certain parameter is set equal to zero or not, will either give you something like relativity's Lorentz transform or will just give you Newtonian physics' Galilei transform, in which there is no time dilation. So you still need experimental confirmation that this parameter is nonzero to show that time dilation will happen, and you also need additional confirmation that this parameter is equal to 1/(speed of light) and thus that the speed of light is the same in all frames.

Also, it's important to remember that one of the assumptions of relativity is that all the laws of physics should work the same way in all inertial frames--if you are in a windowless train car that is moving at constant velocity, then any experiment you do should get the same results regardless of your velocity relative to the earth. If there was any type of clock that did not experience time dilation, this rule would be violated--that's part of the point of the light-clock derivation of the time dilation rule. Brian Greene elaborates on this on pp. 39-40 of The Elegant Universe, after a discussion of light clocks:
The simple but essential point is that the double diagonal path that we see the photon traverse is longer than the straight up-and-down path taken by the photon in the stationary clock; in addition to traversing the up-and-down distance, the photon in the sliding clock must also travel to the right, from our perspective. Moreover, the constancy of the speed of light tells us that the sliding clock's photon travels at exactly the same speed as the stationary clock's photon. But since it must travel farther to achieve one tick it will tick less frequently. This simple argument establishes that the moving light clock, from our perspective, ticks more slowly than the stationary light clock. And since we have agreed that the number of ticks directly reflects how much time has passed, we see that the passage of time has slowed down for the moving clock.

You might wonder whether this merely reflects some special feature of light clocks and would not apply to grandfather clocks or Rolex watches. Would time as measured by these more familiar clocks also slow down? The answer is a resounding yes, as can be seen by an application of the principle of relativity. Let's attach a Rolex watch to the top of each one of the light clocks, and rerun the preceding experiment. As discussed, a stationary light clock and its attached Rolex measure identical time durations, with a billion ticks on the light clock occurring for every one second of elapsed time on the Rolex. But what about the moving light clock and the attached Rolex? Does the rate of ticking on the moving Rolex slow down so that it stays synchronized with the light clock to which it is attached? Well, to make the point most forcefully, imagine that the light clock-Rolex watch combination is moving because it is bolted to the floor of a windowless train compartment gliding along perfectly straight and smooth tracks at constant speed. By the principle of relativity, there is no way for an observer on this train to detect any influence of the train's motion. But if the light clock and the Rolex were to fall out of synchronization, this would be a noticeable influence indeed. And so the moving light clock and its attached Rolex must still measure equal time durations; the Rolex must slow down in exactly the same way that the light clock does. Regardless of brand, type, or construction, clocks that are moving relative to one another record the passage of time at different rates.
So, as long as you suppose that all inertial observers see the laws of physics working the same way, and that this includes Maxwell's laws of electromagnetism which say that the velocity of electromagnetic waves is c regardless of the velocity of the source, then there is no way to avoid these conclusions about time dilation. Of course, it's conceivable one of these premises could be false--in the 19th century physicists believed that Maxwell's laws would only work exactly in a specific preferred reference frame, the rest frame of the "luminiferous ether" which was supposed to be the medium that electromagnetic waves were a vibration in. It was imagined that if our velocity was v relative to the rest frame of the ether, light waves would move at velocity c+v relative to us in one direction, and c-v relative to us in the other. But experiments to detect light travelling at different speeds at different points in the earth's orbit had failed, so this was part of the experimental justification for thinking the assumptions of relativity were true. Since then, all the most fundamental laws of physics that have been discovered have had the mathematical property of "Lorentz-invariance" which insures they will appear the same in all inertial reference frames. Also, there have been experimental observations of time dilation involving things like particles taking longer to decay when their velocity is close to c, and even atomic clocks placed on board the space shuttle which were initially synchronized with clocks on earth but came back a few microseconds behind.

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#### εllipse

In Einstein's Relativity: The Special and General Theory he doesn't mention light clocks once. All of his arguments are based on just using any kind of clock as a measuring device (one assumes an every-day, conventional clock). I'm not sure if Einstein originally used the idea of a light clock to think things through or if the idea of a light clock was invented later by others, but I would guess it was the latter. I don't like the light clock approach (as an introduction to time dilation) for the very reason that it tends to make people think time dilation may only apply to light clocks.

#### selfAdjoint

Staff Emeritus
Gold Member
Dearly Missed
No, Einstein did not use light clocks. They were introduced much later as a pedagogical device, and they work well for that purpose. are you suggesting they are invalid because Einstein didn't use them?

#### εllipse

Not at all. I was merely suggesting they can lead to confusion and are unnecessary because Einstein didn't use them. I think they can be very helpful once you understand time dilation, but shouldn't be used as an introduction to how to understand time dilation. I think conventional clocks should be used first (maybe just in the first chapter or so of SR texts), so that people new to relativity don't assign time dilation to a property of light clocks.

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