Time Dilation Formula: Clarifying Confusion

[JesseM]

Huh? What do you mean when you say "it will take $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ seconds, measured in that same frame, to reach the observer"? First of all, the time dilation formula $$t' = t*\sqrt{1 - v^2/c^2}$$ is normally understood purely in terms of relating the time between ticks in the clock's rest frame to the time between ticks in the frame of an observer moving relative to the clock, the idea that it should have something to do with the time for the light of a tick to reach an observer moving relative to the clock as measured in the clock's own rest frame appears to be an idea unique to you.

If the clock's time is one second, the height is ct where t=1, the horizontal distance is vt' and the diagonal distance is ct', where t' is the time for the light to reach the observer who passed the clock at the start of the 'tick' all measured in the clock's frame, then $${t^'}= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ by the application of simple old Pythagoras.

But to say the observer "passed the clock at the start of the tick" is too vague, this only works if you specifically assume the observer was passing the top mirror at the moment the light was departing from the bottom of the clock; the time to reach the observer would be zero if the observer was passing the bottom at the moment light was departing from there, and somewhere in between zero and the time you give if he was passing the middle. And even if we add in the qualification that you are talking about the time for light from the bottom to reach the observer who passes the top at the moment the light was emitted, I don't really see the point of this calculation--the time for the light from the bottom of the clock to reach the observer has nothing to do with the time the observer will judge for the light clock to make one tick in his own rest frame, and thus nothing to do with the time dilation equation (you could after all place the observer in a completely different position than next to the top mirror when light is emitted from the bottom, in which case the time for the light from the bottom to reach him would be different, but it wouldn't change his judgement about the time of one tick of that light clock in his own frame).

 

[JesseM]

I have been told (not just in this thread) that proper time is the term to use and that there is no such thing as proper time...

Who has told you "there is no such thing as proper time"? It is a very basic idea in relativity, so either you misunderstood what the person was saying, or they are in error.

that there is no need to use A' and B', but that A & B will do: then I am told that I should use A' and B'...

Who said it makes any difference what notation you use for the two frames? All that really matters is that you distinguish them, and explain clearly what each frame is in physical terms (for example, in the time dilation equation it is important to clearly state which of the two frames is the rest frame of the clock whose ticks are being measured, this is usually labeled as the unprimed frame but as long as you clearly explain which frame it is when you write the time dilation equation you are free to use a different notation).

having experienced so much criticism for using the wron terms, I tried defining my own - inertial and transformed units, and earning immediate criticism despite attempting to define exactly what I meant by the use of those terms.

I did not think your explanation of what you meant by these terms was at all clear, but I didn't just criticize, I also asked you to elaborate how the terms would apply to some specific numerical example, in hopes of clarifying.

No matter how I try and ask questions or address the points that don't seem to add up for me all I get is constant critical disection of the language I am trying to use.

Sorry if it seems like you are getting too much criticism, but as I said I've also been asking for clarifications. When discussing a technical subject like relativity, there's no getting around the need for precision in one's use of language but I think this sort of back-and-forth can help make sure we all have a clear idea of what the terms mean, and pinpoint ambiguities that need to be addressed.

One thing which I find particularly annoying (and which I am sure will annoy anyone who experiences it) is to be told what I am thinking, when what I am told is not, and sometimes is the very opposite, of what I am thinking.

I hope I have not "told you what you are thinking", but if the meaning of your words is unclear to me, I think it is helpful to say in my own words what I think you might be saying, so you can respond and tell me something like "yes, that is what I meant" or "no, that's a misunderstanding, let me rephrase". This is all part of the back-and-forth I was talking about...if you didn't know how your words were being interpreted by me, how would we ever figure out if we were on the same page with the meaning of various words and phrases or if we were totally talking past each other?

 

[Rasalhague]

Let me say that I have come to SR on my own using the Einstein paper that I quote from.

I found that to be clear, concise and easy to understand.

Hi Grimble, I don't mean to be snide, but I think you should seriously consider the possibility, if JesseM's explanations seem to you to conflict with this book, that there are flaws in your understanding of what Einstein meant. I know it can be frustrating to think you've got it at last only to be told that you're mistaken. But learning has its ups and downs, and these ideas are notoriously counterintuitive. I'm sure you'd rather know the truth than settle for a superficial feeling of being right.

I then looked further, on the web, principally in Wikipedia etc. and was intersted to find things that did not match what I had learned from Einstein.

Wikipedia can be confusing, especially when a lot of people have worked on one article, and it isn't always right. After all, anyone can contribute to it. If you're trying to learn something new, it can be hard to know how reliable a Wikipedia article is, and the same goes for websites generally. Might I recomment the Relativity chapter in Benjamin Crowel's online physics textbook Simple Nature?

http://www.lightandmatter.com/

A more detailed introduction to special relativity that I've found very useful is Spacetime Physics by Taylor and Wheeler (which isn't online as far as I know). John Baez recommends it in his Guide to Relativity Books.

http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html

I find it helps to read about a difficult topic in a few different textbooks because one often sheds light on aspects of the topic that are unclear in another. Reading different presentations of the same material can also alert me to mistaken ideas I may have formed. Apparent contradictions can sometimes show me that I didn't understand something as well as I thought I did.

I do not pretend (honestly) to know the answers, but for me, just being told 'thats the way it is' doesn't satisfy, I like to know WHY and HOW.

Of course.

One thing which I find particularly annoying (and which I am sure will annoy anyone who experiences it) is to be told what I am thinking,

Sorry if I've added to that frustration by anything I've written! Sometimes I may have said what I thought you meant or suggested possible meanings, but that's only so that you know what meaning I've got from your words, so that you can set me straight if I've misunderstood you.

One of the major problems I find is the constant pulling apart every statement I make and telling me to rephrase it or disecting what the words mean!

Often rephrasing something that people haven't understood can make it clearer. The subject we're talking about involves aspects of reality that are so alien to our everyday experience that we can't rely on our intuition but need to be very careful about our language. Many expressions that make perfect sense in ordinary contexts are imprecise when we're talking about relativity.

 

[JesseM]

Wikipedia can be confusing, especially when a lot of people have worked on one article, and it isn't always right. After all, anyone can contribute to it. If you're trying to learn something new, it can be hard to know how reliable a Wikipedia article is, and the same goes for websites generally. Might I recomment the Relativity chapter in Benjamin Crowel's online physics textbook Simple Nature?

http://www.lightandmatter.com/

A more detailed introduction to special relativity that I've found very useful is Spacetime Physics by Taylor and Wheeler (which isn't online as far as I know). John Baez recommends it in his Guide to Relativity Books.

http://math.ucr.edu/home/baez/physics/Administrivia/rel_booklist.html

Here's another good online intro to SR, written in a Q&A format:

http://www.oberlin.edu/physics/dstyer/Einstein/SRBook.pdf

 

[Grimble]

Hi Grimble, I don't mean to be snide, but I think you should seriously consider the possibility, if JesseM's explanations seem to you to conflict with this book, that there are flaws in your understanding of what Einstein meant. I know it can be frustrating to think you've got it at last only to be told that you're mistaken. But learning has its ups and downs, and these ideas are notoriously counterintuitive. I'm sure you'd rather know the truth than settle for a superficial feeling of being right.

Hello Rasalhague, thank you and I don't think that you are in any way being snide. Your contributions have always been received as couteous and well considered.

Please accept that I agree with your sentiments above! I am quite willing to accept that there may be flaws in my understanding - the possibility of me being right and everyone else wrong is, frankly, not something I would put money on

No, when I say that when I find conflicts, I am trying to convey that the fact that I find conflicts, is something that I need to resolve; my constant carping and saying "but that's wrong" is no help whatsoever! it just encourages the different sides to dig their heels in and pull their hair!

I apologise for this. I should be asking questions not making contraversial statements.

And I am quite happy if someone says "were you thinking ... "
or "it seems to me that what you are saying is ..."
or just to rephrase one of my comments...
That is what would be referred to as "testing understanding" and is a valuable tool in communication...
No, the annoying thing is for someone to say "I know what you are thinking..."
And then go on to restate something far from what I meant or something that I have no problem with.

I am grateful for the time you all spend trying to help an old man.

I will keep at it for I know it will all fit into place, all neat and tidy. It has to for all science is logical, it has to be, that is how it works, we just have to find and understand the right logic.

Your humble student Grimble

 

[JM]

I know that this is a very basic question but what is the correct formula for time dilation?:

The correct formula is the one corresponding to the calculation that you intend to make.
Consider the Lorentz transformations in the form:
x = m ( X - vT), ct = m ( cT - vX/c ),
where m is the term Einstein named 'gamma', y,z = Y,Z = 0, and k( x,t ) is the coordinate frame whose origin moves in the positive direction along the X axis of K( X, T). For the following calculations v = 0.8c, so m = 10/6.
1. Suppose you take Ks point of view ( take K to be at rest) and consider points lying along the line X = vT. Enter this espression in the right side of the transforms to get t = T/m, i.e. t is less than T. Einstein presented this case in his 1905 paper.
2.Take ks viewpoint and consider points along the line x = - vt. Enter this on the left to obtain T = t/m, i.e. t is greater than T.
3. Take ks view and consider points at x = 0 and various values of t. Enter these values on the left and get X = vT, and get T = mt, i.e. T is greater than t. This is the case Einstein presents in the book you cited.
4. Take Ks view and consider points on the line X = 0.5 cT. From the transforms T =t, and x = - X.
From the above we can see that t can be less than, equal to, or greater than T. A single term, such as 'dilation' seems inadequate to describe these varied calculations.

I hope this helps.
JM

 

[Rasalhague]

The angle LaTeX Code: \\beta represents the speed of either frame relative to the other, as a fraction of the speed of light: v/c.

Correction: the angle I labelled beta in these diagrams should have been labelled arctan beta!