Proving the Universality of Physical Laws: A Challenge for Kev

[starthaus]

A scientific formula is right or wrong there is no middle way. I certainly would appreciate it if my formulas or calculations were found wrong, that's the way to learn, you apply, make mistakes, and hopefully someone else takes the trouble of telling you you are right or wrong.

To make things easy let's consider the simplest case a satellite on a circular polar orbit and a clock at one of the poles on Earth. The time ratio is:

\frac{\tau_S}{\tau_E} = 1+\frac{GM}{Rc^2} - \frac{3GM}{2rc^2}

In geometric units it is even simpler:

\frac{\tau_S}{\tau_E} = 1+\frac{M}{R} - \frac{3M}{2r}

G = 6.6726E-11
M = 5.9742E+24 kg (0.004435407 m)
R = 6378000.1 m

So the clock in the satellite is faster if r >\frac{3}{2}R > 3189 km otherwise it is slower.

Yes, this is correct. Here is a more precise (but with same results) calculation:

Start with the Schwarzschild metric for dr=d \theta =0

(cd\tau)^2=\alpha (cdt)^2-(r d\phi)^2

valid for your satellite

and:

the Schwarzschild metric for dr=d \theta =d \phi=0

(cd\tau)^2=\alpha (cdt)^2

valid at the pole

where \alpha=1-2m/r

and m=\frac{GM}{c^2}

\frac{d \tau}{dt}=\sqrt{1-2m/r-(r \omega/c)^2}<1 for any r
The above gives:

\frac{d \tau_s}{d \tau_p}=\sqrt{\frac{1-2m/r-(r \omega/c)^2}{1-2m/R}}

So:

\frac{d \tau_s}{d \tau_p}>1 if

2m/r+(r \omega/c)^2<2m/R

But, from the Kepler's law, it can be shown that :

(r \omega/c)^2=m/r

so, the above becomes:

3m/r<2m/R i.e., your condition r>\frac{3R}{2}

The only problem I see with your proof is that : \frac{3R}{2}=3*6400/2=9600km :-)

Now, GPS satellites are moving at about 20,000km above the Earth, so , if it weren't for the frequency precompensation at launch, their clocks would be faster than the ones left on Earth.

 

[Passionflower]

The only problem I see with your proof is that : \frac{3R}{2}=3*6400/2=9600km :-)

Oops, you are right:

r-R > 3189 km

 

[starthaus]

Oops, you are right:

r-R > 3189 km

Out of curiosity, how did you arrive to your starting formula? As you can see, I had to go through a modest derivation.

 

[Passionflower]

Out of curiosity, how did you arrive to your starting formula? As you can see, I had to go through a modest derivation.

"General Relativity An Introduction For Physicists"
Hobson, Efstathious, Lasenby (Cambridge 2006)

Exercise 7.7

 

[starthaus]

"General Relativity An Introduction For Physicists"
Hobson, Efstathious, Lasenby (Cambridge 2006)

Exercise 7.7

OK, I don't have the book, I derived mine from scratch.

 

[Passionflower]

Reasoning why we have the 3/2R ratio would be very interesting. I wonder if we can relate it to the fact that the photon sphere has the same ratio with respect to the Schwarzschild radius.

Perhaps there is some merit in considering that a stationary observer hovering just above the event horizon has the maximum (coordinate) time dilation due to gravitational time dilation and zero 'SR time dilation' and an observer in a circular orbit slightly outside the photon sphere has the maximum (coordinate and proper) velocity approaching c which implies the maximum 'SR time dilation'. These 'maximums' are 3/2 away from each other.

 

[nismaratwork]

A scientific formula is right or wrong there is no middle way. I certainly would appreciate it if my formulas or calculations were found wrong, that's the way to learn, you apply, make mistakes, and hopefully someone else takes the trouble of telling you you are right or wrong.

Yeah, but not by dragging the thread off-topic into yet another personal pissing match. Do you really think that this is meaningful for the OP, and his question? It's fine to debate, but there is a place for that, and simply blowing out any given thread isn't that place. It's not as if this is the first, or even fifth time either, it's a ****ing pattern that is getting OLD.

Why are the laws of physics the same for all observers, is no longer in the discussion is it? The bottom line, that these are postulates which form the basis for the theory, and not the other way around, is being lost in mathematical minutiae.

Here's an edit, and idea: How about Starthaus and Kev (and you if you like) start a thread where, every time this same old story emerges, you can debate who's right, and how, and why? You'd already have pages of the stuff, and it's a practical way to not drag a given thread into a corner to die quietly.

 

[yuiop]

Here's an edit, and idea: How about Starthaus and Kev (and you if you like) start a thread where, every time this same old story emerges, you can debate who's right, and how, and why? You'd already have pages of the stuff, and it's a practical way to not drag a given thread into a corner to die quietly.

Hi nismaratwork,

when I entered the thread, it appeared to be about clock rates in a gravitational field, because that was what the last half dozen posts was about and I was also responding to comments made about me when I wasn't even in this thread! However, I agree with you that the issues are getting off topic so my apologies for any inconvenience caused.

I have started a new thread for the issues raised by Starthaus/Passionflower/JesseM/myself here: https://www.physicsforums.com/showthread.php?p=2855224#post2855224

 

[nismaratwork]

Hi nismaratwork,

when I entered the thread, it appeared to be about clock rates in a gravitational field, because that was what the last half dozen posts was about and I was also responding to comments made about me when I wasn't even in this thread! However, I agree with you that the issues are getting off topic so my apologies for any inconvenience caused.

I have started a new thread for the issues raised by Starthaus/Passionflower/JesseM/myself here: https://www.physicsforums.com/showthread.php?p=2855224#post2855224

I'm familiar with Starthaus' penchant for calling you out at fairly regular intervals (although I don't know why), and I don't blame you for responding. I think this avenue you're taking now is the best idea. Thanks!

 

[starthaus]

I'm familiar with Starthaus' penchant for calling you out at fairly regular intervals (although I don't know why),

Because kev writes a lot of incorrect things. This is why.

 

[bcrowell]

I'm familiar with Starthaus' penchant for calling you out at fairly regular intervals (although I don't know why), and I don't blame you for responding. I think this avenue you're taking now is the best idea. Thanks!

Kev might be interested in this link: https://www.physicsforums.com/profile.php?do=ignorelist

 

[JesseM]

Look, I am not going to engage in this Q&A game with you again. I know GR quite well and it takes one line of calculations to show that \frac{d\tau}{dt}<1 for any r.

I looked, pervect is using an incorrect metric, his formula for g_{tt} is incorrect.
If you want, I can do the correct one-line calculation.

I looked. kev's calculation has two mistakes:

1. He uses the wrong metric , i.e. wrong g_{tt}

2. even worse, he does not calculate the time dilation for orbital motion

As I suspected, it is alll wrong.

That thread dealt with yet different errors in calculating time dilation for radial (not orbital) motion. No point in bringing into play, do you realize kev's errors in calculating time dilation for orbital motion?

For reference, I've responded to starthaus' various confused criticisms in post #23 on the other thread.

 

[nismaratwork]

Kev might be interested in this link: https://www.physicsforums.com/profile.php?do=ignorelist

He might, but then again he might not want someone spouting off and be unable to directly monitor that, right?

JesseM: I know, I've seen this same pattern in other threads, it's kind of unfortunate. Usually it's a death-knell for the thread, but you're sure as hell not the cause.