Resultant time dilation from both gravity and motion

yuiop

Let's start with this equation for the time dilation ratio:

[tex] \frac{{d}\tau}{{d}t}= \sqrt{1-r_s/r} \sqrt{1- \left (\frac{dr/dt}{c(1-r_s/r)} \right)^2 - \left (\frac{r d\theta/dt}{c \sqrt{1-r_s/r} } \right)^2 - \left(\frac{r\sin\theta d\phi/dt}{c\sqrt{1-r_s/r}} \right)^2 }[/tex]

The above equation is obtained directly from the Schwarzschild metric and I think we are all in agreement about its validity.

Now define local velocities [itex] u_x, u_y, u_z [/itex] as measured by a stationary observer at r using his proper length (dr') and proper time (dt'):

[tex]u_x = dr'/dt' = \frac{dr/dt}{(1-r_s/r)} [/tex]

[tex]u_y = r d\theta/dt' = \frac{r d\theta/dt}{\sqrt{1-r_s/r} } [/tex]

[tex]u_z = r \sin\theta d\phi/dt' = \frac{r \sin \theta d\phi/dt}{\sqrt{1-r_s/r} } [/tex]

Substitute these local velocities into the time dilation equation:

[tex] d\tau/dt = \sqrt{1-r_s/r} \sqrt{1- u_x^2/c^2 - u_y^2/c^2 - u_y^2/c^2 }[/tex]

Now define the local 3 velocity as:

[tex]w = \sqrt{ u_x^2 + u_y^2 + u_y^2 }[/tex]

and substitute this value in:

[tex] d\tau/dt = \sqrt{1-r_s/r} \sqrt{1- w^2/c^2 }[/tex]

This is the result obtained in more detail and more rigorously by DrGreg in #8 and valid for all vertical/horizontal or radial/orbital motion of a test particle at r, just as DrGreg claimed.

I do not think there is anything Starthaus can dispute there.

starthaus

Much better, how did you manage to get the winning combination after all the false starts?
All you needed to do is to start from the correct Schwarzschild metric and to factor out [tex]1-r_s/r[/tex]

JesseM

No, we are not talking about clocks on Earth's surface. You were talking about that in response to Dmitry67's post, not the OP by espen180 (you claim Dmitry67's post was originally on this thread and that it was later split, but I'm not even sure you're correct about that--given that you seemed to think Dmitry67's post was always the first one, it's quite possible the two threads were always separate and that you simply got confused and posted on this thread thinking you were still looking at the other thread, I don't remember Dmitry67's post ever being on this thread). The whole debate between you vs. everyone else got started because you claimed there was something dreadfully wrong with kev's derivation, which was meant to deal with the question of time dilation experienced by clocks in arbitrary circular orbits. The set of all valid circular orbits in Schwarzschild spacetime would include orbits with many different orbital planes, just like Pluto and Earth have different orbital planes despite orbiting the same Sun--some of these orbits would be ones where [tex]d\theta[/tex] is equal to 0 (orbits in the [tex]\theta = \pi/2[/tex] plane), but others would be ones in a different plane where [tex]d\theta[/tex] is not equal to 0. Do you disagree?

starthaus

Yes, look at the post above , kev finally got the right formula after a lot of false starts. Feel free to peruse all his false starts throughout this thread (posts 97, 112, etc). I am done.

JesseM

I haven't been following the more recent posts between you and kev on this thread so I don't know if there were any "false starts" in his last few equations, but the original derivation I linked to was fine, and I pointed out in post 117 you could get the exactly the same formula from the equation you were using in post 6. If you want to use some minor error in a recent post by him as an excuse to avoid addressing all the flaws and confusions in your own arguments (in particular your own confused criticisms of my posts, not kev's), be my guest.

Al68

How about you tell us why instead? Or is this thread destined to exceed 300 posts, too?

Many readers miss out on opportunities to learn due to their understandable unwillingness to sort through 300+ posts of confrontational nonsense that could be avoided by you doing everyone a huge favor by just specifying what you object to and explaining why.

Just try it for once. You might even enjoy being helpful.

starthaus

These are basic math errors, if you can't see them all by yourself, you should find a different hobby. This does not include your current one :trolling.
Enough said that kev understood his errors.

JesseM

If kev made any "basic math errors" in his recent posts to you they are apparently minor ones which don't affect the final equation he derived for circular orbits, as I showed you in post 117 using your own equation to derive it. And of course you have made plenty of minor math errors yourself, like not including the factor of c in post #6.

But never mind, you found a trivial error in someone else's argument, therefore you win the thread! Hooray! (claps very slowly)

Al68

You keep accusing me of trolling, but you are the one who keeps destroying threads, making them completely worthless for most readers, with your shenanigans.

Back to the point, I'll take that as a NO, you either can't or won't explain (in any rational way) your claims.

starthaus

I help people that are sincere, I don't help trolls. You contributted noothing to this thread.
If you can't spot the errors, you have no business (other than trolling) in this thread. But, I'll give you a hint, the errors have to do with the wrong variables in the expression.Compare against the correct final expression.

Al68

LOL. Yep, that's why you can't substantiate your claims: because if I can't "spot the errors" I have "no business in this thread".

Is that also why you're so rude and condescending?

Seriously, dude, it might feel good to say something constructive and useful in your posts.

starthaus

Finally. You understand.


Standard response to your trolling <shrug>


So, you are unable to see the wrong variables. ....
I did, I put up quite a few solutions but you can only see the stuff that makes you tick, err troll.

Al68

LOL. I was responding to your last post which contained nothing useful or constructive.

starthaus

Try reading the other posts, the ones that contain formulas. Can you read formulas?

Al68

I read them. They weren't the ones I was referring to. Was that not obvious? Can you read English?

ZapperZ

Since we are at the point in this thread where people are no longer discussing the topic, but rather who is a troll and who isn't, I take it that the topic is no longer interesting. So this thread is done.

Note that if you think someone is trolling, DO NOT FEED THE TROLL!. Please use the REPORT button and report the post/thread to the Mentors.

Zz.