Same old song and dance - special relativity

[jinksys]

So it's been a year, I've taken differential equations and linear algebra and I'm again enrolled in modern physics. I am also being tormented by the same issues.

1) Could someone post the correct time dilation equation? I have a clock at rest and a clock in motion and I wish to calculate the time elapsed on the clock in motion at a certain velocity.

Is the equation: t' = t / sqrt( 1 - v^2/c^2)

Where t = rest clock, t' = moving clock?

 

[Doc Al]

Is the equation: t' = t / sqrt( 1 - v^2/c^2)

Where t = rest clock, t' = moving clock?

I would switch those around:

Δt = Δt'/sqrt(1 - v^2/c^2)

The meaning is this. During the time that a moving clock shows an elapsed time of Δt', the observing frame would measure a time (on its own 'rest' clocks) of Δt. (Moving clocks run slow.)

 

[yuiop]

So it's been a year, I've taken differential equations and linear algebra and I'm again enrolled in modern physics. I am also being tormented by the same issues.

1) Could someone post the correct time dilation equation? I have a clock at rest and a clock in motion and I wish to calculate the time elapsed on the clock in motion at a certain velocity.

Is the equation: t' = t / sqrt( 1 - v^2/c^2)

Where t = rest clock, t' = moving clock?

t' is the elapsed time between two spatially separated events as measured by two clocks at rest in the observers reference frame. t is the elapsed time as measured by a single clock that is present at both events. It is best not to think in terms of a stationary clock and a moving clock as this can lead to ambiguities and confusion.

For example, let us say we have a light clock that emits a light pulse that is reflected back in one second in the rest frame of the light clock. Now if I am observing the clock moving at 0.8c relative to me, I see the light pulse follow a diagonal path, so the time for the light pulse to return to the emitter takes longer (1.666 seconds) according to my stationary clocks. Now reverse the situation. To the observer at rest with the light clock the tick time is one second. Now if he assumes a moving clock reads a lesser elapsed time, he might conclude that a passing observer will calculate the tick time of his light clock will be less using his moving clocks, but he would be wrong, because we just calculated that the observer moving relative to the light clock measures the tick time to be 1.666 seconds. Therefore considering one clock to be at rest and the other to be moving is ambiguous because either observer can consider themselves to be at rest. Thinking in terms of the proper time (t or t₀) being the time measured by a single clock at both events and coordinate time (t') as being the time measured by two spatially separated clocks, is much better as other posters have pointed out.

[EDIT]I have edited this post to make it a bit less ambiguous.

 

[Zhabka]

Close but I think you have your observed and proper times mixed up. The proper time between two events is that "measured by an observer who see the events occur at the same point in space". So if I am not mistaken (in assuming "the same point in space" means the clock with the proper time is at rest compared to whatever is "ticking"), this means the moving clock will read the proper time, and the dilated time will be measured by a stationary observer.

So for t = time measured by rest clock; t' = time measured by moving clock
the expression would be:

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

With the way you have the expression written you will get time contractions:

Let there be a moving clock (for example some particle moving at 0.9c which decays after one second).

t'=1s

For someone observing this particle the time for decay (using your equation) will be:

1 = t/sqrt( 1 - v^2/c^2)

t = 1*sqrt( 1 - v^2/c^2)
= sqrt(1-0.81)
= 0.436s

The observed time interval is shorter, which it shouldn't be, hence your times were mixed up.

So as long as the observed times get longer and the observed lengths get shorter, you should be on the right track.

 

[jinksys]

The equation I posted was from my uni textbook and is considered correct by my professor. :(

 

[Doc Al]

The equation I posted was from my uni textbook and is considered correct by my professor. :(

What textbook are you using? Ask your professor to show you an example of how that equation is used.

 

[jinksys]

Beiser concepts of modern physics

 

[Doc Al]

Beiser concepts of modern physics

I was afraid of that. I recall having this same issue come up with that very text.

I might have that book in my home library; I'll check it tonight.

Edit: It came up with you a year ago! https://www.physicsforums.com/showthread.php?t=333348

 

[Doc Al]

Beiser's discussion of relativity and time dilation remains one of the worst I've seen.

 

[jinksys]

I asked another professor and he said the equation is right, but that beiser fails to mention that t0 and t are observed by two separate observers.

 

[Doc Al]

I asked another professor and he said the equation is right, but that beiser fails to mention that t0 and t are observed by two separate observers.

The equation that appears in Beiser looks like this:

$$t = \frac{t_0}{\sqrt{1 - \frac{v^2}{c^2}}}$$

There's nothing wrong with that as long as you realize that t₀ is the proper time as recorded on a 'moving' clock and t is the time measured in the frame in which the clock is moving at speed v. An example would be you in your lab measuring the time elapsed on a clock zipping by in some rocket. Most people would call your lab frame the 'rest' frame and the rocket frame the 'moving' frame.

 

[jinksys]

So that equation would give you the time on a clock at rest, given the time elapsed on a moving clock and its velocity, correct?

 

[Doc Al]

So that equation would give you the time on a clock at rest, given the time elapsed on a moving clock and its velocity, correct?

Yes. Rather than get messed up with 'moving' and 'at rest', just consider two frames. There's a clock at rest in one frame that shows an elapsed time of t₀ between two events A and B collocated with that clock. That equation will give the time interval t between those same two events as seen in the other frame--note that in that frame, the events happen at different locations. T₀ is always the proper time as shown on a single clock.

 

[jinksys]

I think it's clicking.

So the collated events could be considered the ticks of the clock in the spaceship, and the ticks would not be at the same location for the t observer as it would for the t0 observer

 

[Doc Al]

So the collated events could be considered the ticks of the clock in the spaceship, and the ticks would not be at the same location for the t observer as it would for the t0 observer

Exactly. In the 'moving' frame of the spaceship, the clock ticks all happen at the same spot since the clock is at rest in that frame. But from the Earth frame, those ticks happen at different locations since the rocket clock moves.

 

[jinksys]

I think I got it, a million thanks!

Since beiser's handling of relativity is so bad, is there a modern physics book you recommend?

 

[Doc Al]

Since beiser's handling of relativity is so bad, is there a modern physics book you recommend?

Assuming Beiser is your required text, I wouldn't necessarily give up on it for other topics. (It's a shame that it has to start off with such a sloppy presentation in chapter 1.) But if you have money to burn, it's always good to have a second text as a backup/different presentation. Something that's confusing in one may be clear in another.

I'm not up on the current crop of Modern Physics textbooks. If you have access to a library, check out what's available. I've seen texts by Serway and Randy Harris that look OK.

If you just want something to solidify your understanding of relativity, there are plenty of web resources. Here's a nice book by Dan Styers: http://www.oberlin.edu/physics/dstyer/Einstein/SRBook.pdf"

 

[jinksys]

Is space-time physics by wheeler still a good guide to relativity?

 

[Doc Al]

Is space-time physics by wheeler still a good guide to relativity?

Absolutely! It's one of my favorites.

 

[jinksys]

I've been reading it for a couple days, it's very good. I like that it doesn't skimp out on the math and at the same time is able to teach the subject clearly.

It sounds like you've read it, so I'll ask you: is this a undergraduate level book on special relativity only? I'm assuming a book on general relativity would be much larger.

 

[Doc Al]

It sounds like you've read it, so I'll ask you: is this a undergraduate level book on special relativity only?

Yes. It only covers special relativity and it is at the undergrad (first or second year) level.

I'm assuming a book on general relativity would be much larger.

You can't judge a book by its length. (OK, you can somewhat.) I have a few pithy books on GR--for example Dirac's book is only 69 pages. (No, I am not recommending that!) But many classic GR books are whoppers--Gravitation by Misner, Thorne, & Wheeler (the same Wheeler) is over 1200 pages.

Taylor and Wheeler is an excellent pedagogical treatment of special relativity. But it doesn't cover every aspect of SR.

 

[jinksys]

I was afraid of that. I recall having this same issue come up with that very text.

I might have that book in my home library; I'll check it tonight.

Edit: It came up with you a year ago! https://www.physicsforums.com/showthread.php?t=333348

Lol yes, hence the title "same old song and dance."