Any way you want is just not good enough. I know that people say that all coordinate systems are equally valid in GR, but they aren't.
Why isn't any way good enough? What criteria are you imposing and where has anyone else applied them?
The picture on the left depicts Schwarzschild coordinates. It takes an infinite coordinate time to cross the event horizon
I already know this. You don't need to post a diagram to prove it to me.
which means it takes forever.
That doesn't follow. First of all, forever according to
who? The Schwarzschild coordinate time does not coincide with what any clock sitting on or crossing the event horizon would measure. On the other hand, a physical clock falling toward the black hole would measure that it reaches the event horizon in
finite proper time.
Adopting Kruskal–Szekeres coordinates is a "humpty-dumpty" way of transforming this away
It is no such thing. The black hole metric in Kruskal coordinates is a perfectly valid solution to the Einstein field equation in its own right. You could prove that without ever knowing about Schwarzschild coordinates. That makes the Kruskal metric a prediction of GR at least as legitimate as the black hole metric in Schwarzschild coordinates.
Now look where that gets you. The Kruskal metric is a) a valid independent solution to the EFE in its own right that's b) mathematically equivalent to the Schwarzschild solution everywhere c) except where the Schwarzschild solution blows up. Given that I'd say you've got no basis for blaming oddities like infinite Schwarzschild time on anything but the Schwarzschild coordinates.
No, I worked this out when I analysed time then gravity and read some original material first-hand. Some people insist that "clocks measure proper time", but actually they don't. Clocks clock up local motion. When you see my light clock going slow due to gravitational time dilation, it isn't because "my proper time is going slower", it's actually my light going slower.
Your light going slower compared with
what?
It is after all a light clock.
Why are you assuming this? GR predicts you'd get exactly the same result with a Casio watch.
That stopped clock means an awful lot, przyk. The Schwarzschild coordinates are giving you a picture of reality, the Kruskal–Szekeres coordinates aren't.
But that's the whole point. Everything you are saying would be true if you could show that the Schwarzschild coordinates were a "picture of reality" (and you could define what you meant by that). But you haven't, and I could give you one or two reasons to believe they aren't. For example, while the time coordinate blows up on the event horizon, no coordinate-independent (read: "physical") quantities do. The Ricci scalar curvature tends to a finite value on the event horizon for instance.
Additionally, you probably believe the Schwarzschild
t coordinate measures time and
r measures distance, but even that isn't generally true. If you read their roles off the Schwarzschild metric, you see the opposite is true near the gravitational singularity:
t becomes a
space-like coordinate there while
r becomes
time-like (incidentally this implies it's much more accurate to think of the singularity at
r = 0 as a point in time rather than in space, which is exactly what you see depicted on a Kruskal chart). The coordinates switch roles right on the event horizon. Essentially there's a "twist" in the coordinate map there. Given that it's hardly surprising you get coordinates blowing up there.
I'll take my cues not just from Einstein, but from the scientific evidence I've referred to. When you see my light clock going slower, just take it face value.
And forget that the light I'm using to see your clock doesn't reach me instantaneously? Forget that there isn't even a well defined notion of "instantaneous" over large distances in GR, which you could use to make the kinds of comparisons you are trying to with spatially separated clocks?
It isn't. The etymological fallacy comes from the "modern interpretation" of GR that isn't in line with Einstein.
No, the etymological fallacy is that you are concluding the geodesic equation in GR must be an equation in space based only on your personal ideas of motion and the fact the phrase "equation of motion" contains the word "motion" in it.
If you really believed that an equation of motion could only be an equation in space, the conclusion you
should draw is that the geodesic equation in GR doesn't fit your definition of "equation of motion" and was mislabeled. Because it specifically
is an equation in spacetime. Again, I've shown you explicitly where Einstein put time-like indices on it and proved that GR would be unable to reproduce Newtonian gravity if it wasn't.
It isn't that. Look at Einstein's operational definition of time.
Sure. Just point me to it.
Clocks clock up local motion, and we call it the time.
Where did Einstein say
that? For starters, saying we use periodic motion to track time doesn't make the motion itself time. Time has other properties associated with it - most notably the notions of simultaneity and synchronicity - that you won't get out of starting with motion as an axiom.
So it's there in his equations of motion.
No it's not. As I told you, the equation explicitly include time-like components.
Pack it in przyk. What I've been saying here is only what Einstein said
No, it's what Einstein said plus the way you choose to interpret it plus your decision to ignore everywhere you've been told why your interpretation (and possibly Einstein's himself) is wrong.
and it's backed up by patent scientific evidence.
Your evidence is invariably both meager (eg. qualitative instead of quantitative) and already accounted for by mainstream theories.
I'm not some "my theory" guy making it up out of fresh air.
No, you're an "I'm following Einstein's legacy" guy. If you don't see what's wrong with that, try substituting "Einstein" for "Jesus" for a minute.
We're talking general relativity here, light is crucial, ice cream is irrelevant. And turning your back on scientific evidence on grounds of interpretation is kicking the scientific method in the teeth.
Apparently you believed the exact opposite when you yourself said "evidence doesn't distinguish between interpretations". In fact I don't know why you're so desperate to back peddle now, because you were right earlier: physical evidence can never distinguish between two interpretations of a theory that use the same mathematics (and therefore make the same predictions).
That's why it's so silly of you to demand I explain things like gravitational time dilation: if you really understood the "modern interpretation" of GR as well as you say you do, you should be able to transliterate your pet explanation for it into "modern" terms yourself.
You're still fighting shy of those light clocks and tape reels. You're parked somewhere near a black hole, you send out two astronauts, each with running tape reels and synchronised light clocks.
Synchronised when and how? When the two astronauts are together before being sent out?
They come back, and their tape-reel readings are different. So where one of them has been, his tape reel was rolling slower.
Er, what exactly are the astronauts trying to measure with their tape reels?
You're employing an axiom that says gravity is curved spacetime, and it's acting as a barrier to a more complete understanding of gravity.
Where do I employ such an axiom? In fact, which of my mathematical proofs are we even talking about here? I posted about three different ones showing three different things and none of them actually had much, if anything, to do with curvature. I certainly never needed curvature as an axiom.
For example, the first was a proof that inhomogeneity of the $$g_{\mu\nu}$$ is coordinate-dependent (and therefore can't be attributed physical significance). The only "axiom" (beyond differential calculus) I used to show this was the transformation rule $$g_{\mu\nu} \rightarrow \frac{\partial x^{\mu}}{\partial x'^{\alpha}} \frac{\partial x^{\nu}}{\partial x'^{\beta}} g_{\mu\nu}$$, which I copied directly out of Einstein's paper.
That's what tells me you can't have followed my derivations. I didn't just prove that inhomogeneity of the $$g_{\mu\nu}$$ was coordinate-dependent based on some axioms I plucked from thin air. I got the sole "axiom" I used from
Einstein. I showed my conclusion was a logical consequence of what
Einstein said about the metric components. (And I put "axiom" in quotes because even Einstein had no choice in the matter. That transformation rule follows from how the metric components are defined.)
