What's non-negotiable is that there is no river of time flowing through an optical clock. So when that clock goes slower it's because the light goes slower. That's it przyk. We got Einstein too, and the Shapiro delay, and the coordinate speed of light varying in a gravitational field, you're ignoring all of it because you've been taught that the speed of light is constant.
This has nothing to do with what I said above. If you tell a story and it depends on you
leaving out information, that's bad science. You are only reinforcing my point with your reply. For example you point out that the coordinate speed of light is variable in GR, but you leave out that:
- it's hopelessly coordinate-dependent,
- it's variable in most noninertial coordinate systems in special relativity, so it is not a new feature of general relativity, and
- the coordinate speed of light is generally not enough information to reconstruct the metric, so you generally cannot explain all gravitational effects just in terms of the coordinate speed of light.
(The last point is obvious just by parameter counting: the speed of light is just one function. The coordinate velocities in the
x,
y, and
z directions gets you up to three functions. That's not enough information to recover the
ten independent $$g_{\mu\nu}$$s. So you can't throw away the $$g_{\mu\nu}$$s and pretend it's only the coordinate speed of light that you need to know. You lose information about the gravitational field if you do that.)
I don't. I dismiss it as quote mining because it doesn't tally with how Einstein actually formulated general relativity in 1915-1916 and it doesn't tally with how detailed predictions are made from the theory (a topic that you consistently avoid like a plague).
You dismiss
what Einstein said and all the evidence because it doesn't tally with what you've been taught.
No, I am dismissing your quote mining for exactly the reason you quote me saying above. Instead of addressing that intelligently you have replied with an ad hominem.
See what I said about the optical clock above and replace t by c, like this:
$$c_0 = c_f \sqrt{1 - \frac{2GM}{rc^2}}$$
But then you've got a problem because of the c in the $$rc^2$$. Reformulating GR to give the God's eye view instead of the local view isn't easy. Again see
http://arxiv.org/abs/0705.4507 along with
http://arxiv.org/abs/astro-ph/0703751 where Ellis said
"any proposed variation of the speed of light has major consequences for almost all physics". Magueijo and Moffat said this:
"As correctly pointed out by Ellis, within the current protocol for measuring time and space the answer is no. The unit of time is defined by an oscillating system or the frequency of an atomic transition, and the unit of space is defined in terms of the distance travelled by light in the unit of time. We therefore have a situation akin to saying that the speed of light is “one light-year per year”, i.e. its constancy has become a tautology or a definition".
I appreciate that you're trying here, but you're missing the point. If I just want to work out the coordinate speed of light I already know how to do that easily. Einstein explains that in
section E.22 of his 1916 paper, for instance:
Albert Einstein said:
We now examine the course of light-rays in the static gravitational field. By the special theory of relativity the velocity of light is given by the equation
$$-\, dx_{1}^{2} \,-\, dx_{2}^{2} \,-\, dx_{3}^{2} \,+\, dx_{4}^{2} \,=\, 0$$
and therefore by the general theory of relativity by the equation
$$ds^{2} \,=\, g_{\mu\nu} dx_{\mu} dx_{\nu} \,=\, 0 \qquad (73)$$
If the direction, i.e. the ratio $$dx_{1} \,:\, dx_{2} \,:\, dx_{3}$$ is given, equation (73) gives the quantities
$$\frac{dx_{1}}{dx_{4}},\ \frac{dx_{2}}{dx_{4}},\ \frac{dx_{3}}{dx_{4}}$$
and accordingly the velocity
$$\sqrt{\Bigl(\frac{dx_{1}}{dx_{4}}\Bigr)^{2} \,+ \, \Bigl(\frac{dx_{2}}{dx_{4}}\Bigr)^{2} \,+\, \Bigl(\frac{dx_{3}}{dx_{4}}\Bigr)^{2}} \,=\, \gamma$$
defined in the sense of Euclidean geometry.
Apply that for the weak field metric I wrote in [POST=3169212]this post[/POST] and you'll get $$\gamma \,=\, c \bigl( 1 \,+\, \Phi / c^{2} \bigr)$$, where $$\Phi$$ is the Newtonian gravitational potential, for instance. Do it with the Schwarzschild metric and you'll get the (radial) coordinate speed $$\gamma \,=\, c \Bigl( 1 \,-\, \frac{2GM}{r c^{2}} \Bigr)$$.
That's easy. The
problem is that this is all explicitly coordinate dependent, through and through. Einstein even points this out in the very next sentence with regard to the "bending" of light (emphasis added):
Albert Einstein said:
We easily recognize that the course of the light-rays must be bent with regard to the system of co-ordinates, if the $$g_{\mu\nu}$$ are not constant.
More problematic for you is that the coordinate speed of light alone generally isn't enough information to predict that bending (Einstein only considers the special case of a static gravitational field in the section I'm quoting from). You need the geodesic equation for that. Einstein explains this in
section C.13. He gives the reasoning for a material point, but the same argument works for a light ray:
Albert Einstein said:
A freely movable body not subjected to external forces moves, according to the special theory of relativity, in a straight line and uniformly. This is also the case, according to the general theory of relativity, for a part of four-dimensional space in which the system of co-ordinates K[sub]0[/sub] may be, and is, so chosen that they have the special constant values given in (4).
If we consider precisely this movement from any chosen system of co-ordinates K[sub]1[/sub], the body, observed from K[sub]1[/sub], moves, according to the considerations in § 2, in a gravitational field. The law of motion with respect to K[sub]1[/sub] results without difficulty from the following consideration. With respect to K[sub]0[/sub] the law of motion corresponds to a four-dimensional straight line, i.e. to a geodetic line. Now since the geodetic line is defined independently of the system of reference, its equations will also be the equation of motion of the material point with respect to K[sub]1[/sub]; If we set
$$\Gamma_{\mu\nu}^{\tau} \,=\, -\, \Bigl\{ {\mu\nu\atop \tau} \Bigr\} \qquad (45)$$
the equation of the motion of the point with respect to K[sub]1[/sub], becomes
$$\frac{d^{2}x_{\tau}}{ds^{2}} \,=\, \Gamma_{\mu\nu}^{\tau} \, \frac{dx_{\nu}}{ds} \, \frac{dx_{\nu}}{ds} \qquad (46)$$
We now make the assumption, which readily suggests itself, that this covariant system of equations also defines the motion of the point in the gravitational field in the case when there is no system of reference K[sub]0[/sub], with respect to which the special theory of relativity holds good in a finite region. We have all the more justification for this assumption as (46) contains only the
first derivatives of the $$g_{\mu\nu}$$, between which even in the special case of the existence of K[sub]0[/sub], no relations subsist.
Basically, if you want to derive the equation of motion for a material point or light ray in some coordinate system K[sub]1[/sub], you do it applying the following logic:
- Switch to a locally inertial coordinate system K[sub]0[/sub] (defined as one in which the metric becomes the Minkowski metric and all the metric gradients vanish at the point in space and time under consideration).
- Assert that the material point or light ray follows a straight world line (i.e. obeys Newton's first law) in K[sub]0[/sub].
- Transform back to the coordinate system K[sub]1[/sub].
So the "bending" of worldlines is derived in general relativity, in an
explicitly coordinate-dependent way, from the assumption that the worldline is locally
straight in a locally inertial coordinate system. Of course, it should go without saying that if you can make the "bending" of a trajectory zero or non-zero just by changing the coordinates, it is
not a very remarkable physical quantity.
As far as I know, there is only really one
invariant measure of anything like "bending" in general relativity, though it's not discussed by Einstein: the relative convergence or divergence of neighbouring worldlines. This is given by the
Jacobi equation, which relates the convergence or divergence of infinitesimally separated geodesics to the Riemann curvature tensor. So the
one really invariant measure of "bending" of trajectories in general relativity is explicitly dependent on the one quantity whose importance you keep trying to downplay: the spacetime curvature.
Finally, the appearance of the constant
c is not and never has been a problem in general relativity. It's just the speed of light you'd measure under locally inertial conditions with local measurements, regardless of the details of how you define your system of units. Since 1983 (and
only since then, long after either theory of relativity was first formulated) it's also been adopted as the official definition of how the metre is scaled relative to the second, based on the
presumption that it should be locally invariant anyway.
[NB: the quotations from Einstein's 1916 paper above are as they appear in the translation
The Collected Papers of Albert Einstein, Vol. 6,
The Berlin Years: Writings, 1914-1917, Doc. 30.]
The lower clock ticks slower, stop this ducking and diving.
You've already admitted that isn't true for clocks on the ISS. You even linked to a graph illustrating that!
I didn't. The situation started with you suspended, then we let you go. Then you were in free fall. And you are testing my patience.
No, this sub-discussion started with this comment I made in [POST=3169800]post #177[/POST]:
Concerning point 2), you conveniently ignore that, according to general relativity, gravitational time dilation effects are predicted to largely disappear under free-fall conditions. You certainly can't challenge this on experimental grounds, because that experiment has never been performed.
If that wasn't explicit enough, after a comment by OnlyMe I made it unambiguously clear what sort of situation I was referring to in [POST=3169815]post #179[/POST] (emphasis added):
The sort of situation I was thinking of would be more like doing the NIST optical clock experiment on the International Space Station or (if it could somehow be done accurately and quickly enough) on a
reduced gravity aircraft,
with both clocks approximately in free-fall and kept at some fixed distance from one another.
Since you had yourself remaining outside the event horizion instead of in free fall, you haven't addressed the point I was making.
It is true. We all understand gravitational time dilation. Clocks run slower when they're lower. That's it. Stop trying to suggest it isn't true.
You sound like you don't like having your beliefs about general relativity challenged.
Not so. See above. And note what Einstein said. It's clear that you don't care what the guy said.
As a rational thinking person, I care about what Einstein said to the extent, and
only to the extent, that he was able to
justify what he said. So if you quote Einstein saying something that isn't accompanied by a derivation, and your interpretation of it
contradicts what Einstein said in a work like his 1916 paper where he gave
very detailed derivations, you can certainly take it for granted I won't be impressed.
