Time Dilation

[tommyedison]

In the special relativity theory, Einstein uses the train-platform example to calculate the formula for time dilation in which the person inside the train fires a vertical light beam which appears slanted to the observer on the platform. Let's take an experiment in which the observer inside the train, instead of firing a vertical beam of light, fires a slanted beam of light which is reflected from the mirror on the ceiling. The beam then will appear even more slanted to the observer outside. I then tried to derive the equation for time dilation through this example using sort of the same method which is generally used. However my equation did not match the time dilation equation. Where did I go wrong? Did I go wrong on the concept?

Also, I would like to know Theory of Elementary Waves' position on time dilation.

[stephen_speicher]

tommyedison, I do not think that learning relativity from Einstein's non-technical book is a good way to learn relativity nowadays. From your description it is hard for me to fathom just what you are trying to do, and I have seen many people stumble unnecessarily over relativity because of the wrong sources. I would strongly suggest that you try Spacetime Physics, Edwin F. Taylor and John Archibald Wheeler, W.H. Freeman and Company, 1992, for an excellent non-technical introduction to the subject. I think you will find it much more easy to follow.

As to time dilation: It is a well-defined and sensible concept that has been experimentally demonstrated countless times.

[Capitalism Forever]

As to time dilation: It is a well-defined and sensible concept

What exactly is its definition?

[tommyedison]

What exactly is its definition?

Time dilation says that relative motion changes the speed at which time passes. This means that if a person is travelling at high speed and another person is standing still, then from the reference frame of the person who is standing still, time will pass more slowly for the person who is moving at high speed.

[stephen_speicher]

What exactly is its definition?

In special relativity time dilation is essentially the result of a measurement process. For two observers in uniform motion relative to each other, each observer will measure the tick rate of the other's clock to be slower than his own.

[stephen_speicher]

Time dilation says that relative motion changes the speed at which time passes. This means that if a person is travelling at high speed and another person is standing still, then from the reference frame of the person who is standing still, time will pass more slowly for the person who is moving at high speed.

This is why I suggested reading a good conceptual introduction to special relativity, such as Spacetime Physics. The above expanation gives the impression that time is something which is flowing and that somehow it speeds up or slows down. For each of the two observers their clocks tick at the same rate in their own reference frames; there is no change to "the speed at which time passes" as measured by their own clocks. This is an all too common misunderstanding in most popularizations of relativity, even so in many beginning texts.

[Spearmint]

Steven, could you recommend a particular edition of Spacetime Phyiscs please? I got one out of my local library and haven't started reading it yet but it seems a bit difficult to follow - theres loads of boxouts everywhere, and my initial impression was that it was kind of 'scattered'. Just wondering if this is the one you recommend - if it is, then I'll dive into it anyway and find out what it's like; its probably that I'm just not used to that particular style of writing/presentation.

Ta.

[stephen_speicher]

Steven, could you recommend a particular edition of Spacetime Phyiscs please? I got one out of my local library and haven't started reading it yet but it seems a bit difficult to follow - theres loads of boxouts everywhere, and my initial impression was that it was kind of 'scattered'. Just wondering if this is the one you recommend - if it is, then I'll dive into it anyway and find out what it's like; its probably that I'm just not used to that particular style of writing/presentation.

Ta.

The original edition was from the 1960s, and it is a little less precise in formulation of concepts but it has slightly more math and more worked examples. It is usually available through used book sources.

The newer edition is from the 1990s, and that is the one mostly used. The "boxes" are a style that was perfected in the "bible" for the field, Gravitation by Misner, Thorne, and Wheeler, an almost 1300 page tome. The style was carried over in the newer edition of Spacetime Physics by Wheeler. Perhaps it takes a bit of getting used to, but it is well worth it.

[Spearmint]

Ok, thanks. I'll stick with the one I have in that case.

[Capitalism Forever]

In special relativity time dilation is essentially the result of a measurement process. For two observers in uniform motion relative to each other, each observer will measure the tick rate of the other's clock to be slower than his own.

I see. But the observer can adjust his measurement for time dilation and that way he can determine how fast the other's clock really goes--is that right?

[stephen_speicher]

I see. But the observer can adjust his measurement for time dilation and that way he can determine how fast the other's clock really goes--is that right?

There are explicit formulas which encapsulate relativistic relationships, so given that the measured quantities are known (relative speed, tick rate, etc.) then an observer in relative motion to another can calculate the same tick rate of the clock as done by the observer for whom the clock is at rest.

But, I would be careful about what is meant by your statement "really goes." All measurements we make are 'real' and are not distortions; objects we measure do not themselves change but what we mean by the length of an object or the elapsed time between events is a physical process in which we use our rulers and clocks. Relativity singles out the notion of 'proper length' and 'proper time' as quantities measured by an observer at rest with the object, but that does not mean that such a measurement is more real than a measurement made by an observer in relative motion to the object. The object exists in objective reality but these reference frames are of our own creation.

With that said, a similar idea is used in the much more complex Global Positioning System (GPS). There are atomic clocks in the various orbiting satellites of the system and time measurements are affected by a variety of relativistic effects that are much more sophisticated than simply time dilation. These various effects are calculated according to general relativity and the frequency of the orbiting clocks are preadjusted at the factory to account for the many relativistic effects. That way the different orbiting clocks are synchronized with the master atomic clock here on Earth. The GPS system is a magnificent tribute to general relativity, without which the entire system would not work.

[Bryan]

QUOTE (stephen_speicher @ Jul 21 2004, 10:09 PM)

In special relativity time dilation is essentially the result of a measurement process. For two observers in uniform motion relative to each other, each observer will measure the tick rate of the other's clock to be slower than his own.

I see. But the observer can adjust his measurement for time dilation and that way he can determine how fast the other's clock really goes--is that right?

The observer can calculate for time dilation based the speed of the other's inertial frame.

The Theory of Special Relativity is based on Einstein's two postulates:

1. The speed of light is the same for all observers, regardless of their inertial frame. (The speed of light is the only thing that remains constant)

2. The laws of physics are the same in any inertial frame of reference.

Based on the above two postulates, time dilatation can be calculated by the following equation:

t' = t * sqrt(1-(v/c)^2)

t' is the dialated time, t is the measured time in the observing inertial frame, v is the velocity of the "other" inertial frame and c is the speed of light.

How fast "other's clock really goes" is kind of an arbitrary statement because its different depending on your own inertial frame.

[stephen_speicher]

The Theory of Special Relativity is based on Einstein's two postulates:

Plus several "hidden postulates" that Einstein explicated later.

1.  The speed of light is the same for all observers, regardless of their inertial frame.  (The speed of light is the only thing that remains constant)

2.  The laws of physics are the same in any inertial frame of reference.

There are several errors here.

Your "1." is not what Einstein postulated, but rather a conclusion drawn from combining Einstein's actual first and second postulates. Here are the two main postulates (Note also that you reversed Einstein's order):

"1. The laws governing the changes of state of any physical system do not depend on which one of the two coordinate systems in uniform translational motion relative to each other these changes of the state are referred to.

"2. Each ray of light moves in the coordinate system 'at rest' with the definite velocity V independent of whether this ray of light is emitted by a body at rest or a body in motion."

(Albert Einstein, On the Electrodynamics of Moving Bodies, Annalen der Physik, 17, 1905.)

So Einstein's actual second postulate only applies in the single frame specified, and it is only by combining his second postulate with the first postulate, the "principle of relativity," that one can state, as you did in your supposed postulate, in regard to light speed being "the same for all observers, regardless of their inertial frame."

Also, your remark that "The speed of light is the only thing that remains constant" is incorrect on two counts. First, you do not mean 'constant' in this context, but rather 'invariant.' A constant is something that does not change with time, and an invariant in relativity is a quantity that has the same value regardless of in which inertial frame it is measured. Second, the speed of light is certainly not the only invariant in special relativity; there is the invariant interval, for instance, or invariant mass for another.

Also, your description of the time dilation equation is a bit confusing, but I do not want to take the time to detail the derivation and explanation. Perhaps another time if it comes up in some other context.

[Bryan]

There are several errors here.

I'll concede that I messed up the order and the wording of the postulates as I recited them from memory based on an undergraduate modern physics course I took last spring, but the implications are the same. Do a google on "Einstein's postulates," each summary has them worded differently but they every meaning is essentially the same.

Also, your remark that "The speed of light is the only thing that remains constant" is incorrect on two counts. First, you do not mean 'constant' in this context, but rather 'invariant.' A constant is something that does not change with time, and an invariant in relativity is a quantity that has the same value regardless of in which inertial frame it is measured. Second, the speed of light is certainly not the only invariant in special relativity; there is the invariant interval, for instance, or invariant mass for another.

Einstein's words:

"The second principle, on which the special theory of relativity rests, is the 'principle of the constant velocity of light in vacuo.' This principle asserts that light in vacuo always has a definite velocity of propagation (independent of the state of motion of the observer or of the source of the light)."

(Albert Einstein, What is the Theory of Relativity?, in Ideas and Opinions)

By my definition, and apparently Einstein's too, 'constant' and 'invariant' are interchangeable terms, the term 'constant' is not coupled with time. Again my knowledge is based on an undergraduate modern physics course, but it is my understanding that the speed of light is the only constant (or invariant, if you wish) in special relativity. An interval is either the space between two points or a specified amount of time, how do these remain the same in any inertial frame?

Also, I thought that mass was relative and explained by this equation:

m' = m*(1-(v/c)^2)^-(1/2)

Where m' is the relativistic mass of an object in motion, m is the rest mass of the object, v is the velocity of the object, and c is the speed of light.

Can you give an example of when a mass would be invariant?

Also, your description of the time dilation equation is a bit confusing

I agree that the equation is confusing but that is only because you cannot type square root symbols and exponentials in plain text. Nevertheless, the equation is correct.

[stephen_speicher]

I'll concede that I messed up the order and the wording of the postulates as I recited them from memory based on an undergraduate modern physics course I took last spring, but the implications are the same.  Do a google on "Einstein's postulates," each summary has them worded differently but they every meaning is essentially the same.

Bryan, I understand that you are repeating what you may have learned in your "undergraduate modern physics course," and I am well-aware of what one may find by searching google. But what you may been taught, and what others write on the internet, is a result of a certain standard of scholarship, not a standard itself. You have missed my point in regard to the postulates.

I gave you a proper translation of Einstein's two postulates from his 1905 paper on electrodynamics, and if you go back and carefully read what I wrote I would hope that you see what you attributed as being a postulate was not advanced by Einstein at all. What you stated as a postulate is an inference drawn from the two actual postulates that Einstein presented. In many beginning courses, and in many beginning texts, and in many popularizations, and on many web pages, there are many examples of loosely formulated ideas that represent poor scholarship and bad conceptualization. I presented you with the facts, and I explained how to interpret them, so it is up to you as what conclusions you draw.

As to the concept of an invariant in special relativity, it has a very precise technical meaning and it is most definitely different from what is termed a constant. In physics and elsewhere I have a great respect for the precise meaning of terms, and I explained the essential difference between invariant and constant. You are free to ignore what I wrote if you like, but if you do I caution that you will be continuing to fill your mind with imprecise notions. I take my physics quite seriously and I strive to define and use my terms with the same care and precision that is required for philosophy. If you study mathematics and physics at a a sufficiently high-enough level, you will find, in general, that there is a much greater respect for precision than you will find in your beginning undergraduate work.

Again my knowledge is based on an undergraduate modern physics course, but it is my understanding that the speed of light is the only constant (or invariant, if you wish) in special relativity.  An interval is either the space between two points or a specified amount of time, how do these remain the same in any inertial frame?

The invariant interval is a fundamental concept in special relativity. It represents the objective facts that all observers will agree upon, regardless of their relative uniform motion. Due to the relativity of simultaneity, the measured spatial separation between two events, and the measured time separation between those same events, will, in general, vary among observers in relative motion to each other. The particular mix, if you will, of spatial and time components are different between observers. But, there is an objective relationship, the invariant interval, which will always be the same for all observers, regardless of their spatial and time mixture. Simply put, the relationship is:

(invariant interval)^2 = (c * time separation)^2 - (space separation)^2.

This invariant interval is a fundamental of special relativity, and if it was not explained as such you have unfortunately missed out on a key premise of the theory.

Also, I thought that mass was relative and explained by this equation:

m' = m*(1-(v/c)^2)^-(1/2)

Where m' is the relativistic mass of an object in motion, m is the rest mass of the object, v is the velocity of the object, and c is the speed of light.

Can you give an example of when a mass would be invariant?

This too is a poorly taught and poorly understood concept on the beginning level, and I do not blame you for the confusion you might have. In modern physics mass is an invariant quantity, and when a particle physicist uses the unadorned term mass he is referring to the norm of the momentum 4-vector, a quantity that is independent of coordinate system. The mass is given by the general relativistic formulation, m^2 = (E^2 - C^2p^2)/c^4, where E is the energy and p is the norm of the 3-momentum.

The "relativistic mass" that you refer to is an archaic formulation that was essentially discarded some time ago. Relativistic mass is just the 4-component of the energy-momentum vector, which is just energy. Energy is NOT an invariant quantity, but the invariant mass is independent of motion and coordinate system.

I grant that you will not usually learn these facts in popularizations and in many beginning courses, but if you study advanced texts these issues will be made clear, especially the technical reasons which underly the formulations. Also, if you read the actual physics journals you will see that, for instance, almost always when particle physicists refer to the unadorned term 'mass' they are referring to an invariant quantity.

But even at the beginning level you can learn the proper conceptual foundation for special relativity, which is why I have been so strongly recommending to the non-technical people Taylor and Wheeler's Spacetime Physics. For reference, another excellent non-technical book is General Relativity From A to B," Robert Geroch, The University of Chicago Press. The main sections of the book have a wonderful introduction to special relativity, and the ending sections introduce general relativity concepts on a very basic level.

[Capitalism Forever]

But, I would be careful about what is meant by your statement "really goes." All measurements we make are 'real' and are not distortions; objects we measure do not themselves change[...]

Yes, that's what I meant by "really" : that the objects we measure do not themselves change, only the measurements. "Really," as contrasted with "apparently," as in: "That stick immersed in water isn't really bent; it just appears so because of the refraction of light."

By the same token, "That clock isn't really slow; it just appears so because of time dilation." Do you think it would be correct to say that?

[Tom Rexton]

Yes, that's what I meant by "really" : that the objects we measure do not themselves change, only the measurements. "Really," as contrasted with "apparently," as in: "That stick immersed in water isn't really bent; it just appears so because of the refraction of light."

By the same token, "That clock isn't really slow; it just appears so because of time dilation." Do you think it would be correct to say that?

Mr. Speicher said "For two observers in uniform motion relative to each other, each observer will measure the tick rate of the other's clock to be slower than his own."

Your question seems to imply that both observers "really" have the same clock tick-rate--that neither clock is "really" slower than the other, only appears to be so because of their measurements.

Isn't that presupposing the existence of a universal, uniform time-rate which is the "real" as opposed to the "measured" rate of both clocks?

[stephen_speicher]

Yes, that's what I meant by "really" : that the objects we measure do not themselves change, only the measurements. "Really," as contrasted with "apparently," as in: "That stick immersed in water isn't really bent; it just appears so because of the refraction of light."

But there is an essential difference between time dilation and the stick-in-the-water scenario. Let me switch from time dilation to length contraction to more easily illustrate. Like time dilation, length contraction for two observers in relative uniform motion to each other means that each will measure the others standard rulers to be shorter than their own.

The stick in the water appearing bent is strictly a perceptual issue, one in which we directly experience the evidence of the senses. If all we had to go by was direct perception, we would just see the stick bent in water, and straight otherwise. We would not have any basis to say whether or not the stick was "really" straight or "really" bent. Which would be the "appearance," and which would be the "reality?" It is only by reference to some other process -- some means of measurement and inference -- that we ultimately learn that the "natural" state of the stick is being straight, and it only appears to be bent in the water.

But with length contraction we are not talking about direct perception. An "observer" in special relativity does not have the same commonplace meaning that "observation" plays in direct perception. In relativity an observer stands for a measurement process, not what one sees with one's eyes. An observer is a whole array of synchronized clocks and standard rulers spread out through one's inertial reference frame, each capable of recording events in its vicinity and reporting the results back to a central place.

What we "see" with our eyes is different from what is measured by an observer. When we refer to an "observation" of an object made with one's eyes, we mean the entire collection of light rays which converge upon one's eyes. The use of "observation" in this manner is completely different from the meaning of "observer" as I described above. If we were here concerned with direct perception, with what we see with our eyes, then this would lead us to the relativistic Doppler effect, a different story entirely.

So for the observer length contraction is a measurement process, not direct visual perception. What we mean by length is the simultaneous measurement of an object's end-points, and because of the relativity of simultaneity these measurements will differ for different observers in relative motion to each other. This is a physical difference tied to the very process of measurement -- which itself is essentially dependent on the light we ultimately use -- and the length of the object for different observers is physically different. This does not mean that the object itself changes for different observers, but what we mean by length physically changes for different observers due to the nature of the only process we have for measuring reality.

[Capitalism Forever]

Isn't that presupposing the existence of a universal, uniform time-rate which is the "real" as opposed to the "measured" rate of both clocks?

No, it just presupposes the existence of a post-adjustment tick rate as opposed to a pre-adjustment tick rate. In other words, by "really," I mean "after adjustment," or "discounting the effects of my means of observation."

"The clock looks slow to me, but that is just because I am observing it while moving relative to it. Eliminating the effect of my motion upon my observations, I can conclude that the clock goes just as fast as it should; the guy traveling with the clock can use it as a reliable means of chronometry."

It's a bit like doing exchange rates. "When I was in Sydney, I paid $20 for dinner. However, it appears as $15 on the statement from my credit card company. But they didn't really undercharge me because the restaurant bill was in Australian dollars and the credit card statement is in U.S. dollars. After adjustment for the exchange rate by converting to Australian dollars to U.S. dollars, the restaurant bill says $15, which is exactly what was charged on my credit card."

[Capitalism Forever]

An "observer" in special relativity does not have the same commonplace meaning that "observation" plays in direct perception. In relativity an observer stands for a measurement process, not what one sees with one's eyes. An observer is a whole array of synchronized clocks and standard rulers spread out through one's inertial reference frame, each capable of recording events in its vicinity and reporting the results back to a central place.

I see. (No pun intended. ) This is quite an important point to clarify because when I hear "observer," I automatically tend to think of an "eye-observer," i.e. a single point where rays of light coming from different directions are intercepted and processed. The point you have made above clears up pretty much confusion I have had about frames of reference.

what we mean by length physically changes for different observers due to the nature of the only process we have for measuring reality.

I think this is quite analogous to my currency example in my reply to Tom above. Sums of money can only be expressed in terms of a specific currency. When you don't specify the exact currency, your sum is relative; $20 means something different in Australia than what it means in America. But when you say "U.S.$20," you have an absolute sum. There is a translation between amounts expressed in terms of different currencies, e.g. "U.S.$20=AUS$15" and there is no single "real" currency--but when you adjust for the exchange rate, you are able to compare prices in different countries. An item costs $32 in Australia and $28 in the States--but is it really cheaper in the States?

Once you single out (arbitrarily, by necessity) a currency in terms of which you will express how much the item "really" costs, you can use that to make comparisons.

Similarly, you can single out each clock's own frame of reference for expressing the "proper" tick rate of the clock, and that way you can tell whether or not the clocks have been calibrated to go at the same speed.

In the end, it all boils down to the fact that, while all knowledge is contextual, reality itself is independent of context; you express facts of reality in terms of a certain context, but once you have fixed your context--your currency, your time zone, your frame of reference--your expressions refer to absolute facts of reality. You can translate it to another currency/time zone/frame of reference, and the translated version will have different figures in it, but it will still refer to the same facts of reality.

Would you agree with the above?

[stephen_speicher]

... Would you agree with the above?

I'm sorry, but I am having difficulty following your analogy(ies) and seeing its relevance. I am also unclear about the precise meaning of your use of several words and phrases. I would rather stay directly with the actual subject matter at hand and use the words and ideas that I carefully explained.

[Tom Rexton]

I don't think the currency exchange rate analogy applies well to time dilation.

Using the equation 20 USD = 15 AUD to determine if 30 AUD is "really" cheaper than 35 USD is simply a conversion of different monetary units.

In time dilation, the measurements of the clocks' tick rate differ even though both are measured in seconds.

Say Observer A measures the following: 1 second of A's clock = 0.5 second of B's clock.

Observer B measures the opposite: 1 second of B's clock = 0.5 second of A's clock.

How would you adjust either to the "proper" rate? Who's time is "really" slower? I don't think that's a question one can ask, because, if I'm not mistaken, according to relativity there is no rate of time independent of any reference frame. Asking the question "is his rate of time slower?" begs the question "from what frame of reference?"

[Capitalism Forever]

I'm sorry, but I am having difficulty following your analogy(ies)

I am sure you are capable of following an example of currency conversion, so I take it you're not motivated to talk about exchange rates on a science thread...which is perfectly understandable.

Consider the following scenario, then: A spaceship with an astronaut in it is moving away from Mission Control in Houst...um, wherever our private space exploration company is headquatered, at a uniform speed v in a straight line. A device for measuring his pulse emits a radio signal every time his heart beats. His doctor, who sits at Mission Control and monitors the signals, will observe n signals per minute. He adjusts this for the "redshift" resulting from the fact that the spaceship is moving away, and obtains n'. This n' is still different from the pulse rate he measured for the same astronaut when he examined the astronaut on Earth before liftoff. Should the doctor conclude that the heart of the astronaut has slowed down, possibly due to some medical condition?

I would say that he shouldn't. He should calculate the time dilation factor corresponding to v and multiply n' with that factor to obtain n'', which is the astronaut's proper pulse rate, and compare that to what he measured before liftoff.

Am I right about this?

[Capitalism Forever]

Asking the question "is his rate of time slower?" begs the question "from what frame of reference?"

"Is his tick rate, in his frame of reference, slower than my tick rate in my frame of reference?"

[dsandin]

"Is his tick rate, in his frame of reference, slower than my tick rate in my frame of reference?"

I think something has been lost here, for this question even to come up. Your "tick rate" comparision question is founded on the differences between frames (their relative velocity and the symmetry between observed unequal clock readings). But at root, by "tick rate" in your question you actually are getting at the amount of change inherent in a unit of the clock's action. The principle of special relativity -- which affirms that the objective physical reality of an event is not dependent on the inertial frame to which measurements of it are referred -- is all you need in order to realize in the context of your question that the "tick rates" of identical clocks are identical, independent of frame. Indeed, the law of identity ought to suffice.