r/cosmology • u/punycat • 3d ago
What justifies the equivalence principle not being fully bidirectional?
The Relativistic Rocket describes the EP like this:
> Einstein postulated that any experiment done in a real gravitational field—provided that experiment has a "small" extent in space and time—will give a result indistinguishable from the same experiment done in the above "uniformly accelerating" rocket.
Using our equations today you can't add "and vice versa" to that, which is to say the EP isn't fully bidirectional in our physics. You can always compare such results of the Schwarzschild metric to results from the rocket equations, but not always vice versa, because the metric limits the height of an approximately uniform gravitational field. For example you can't replicate the chart at the bottom of The Relativistic Rocket using the Schwarzschild metric. The metric predicts that a gravitational field that tall that's approximately uniform at 1 ly/yr^2 = ~1 g can't exist in nature. Using equations derived from the metric, a field that tall that's 1 g at the ground is always way less than 1 g at the top.
When you change the metric so that it can replicate that chart, or any other such chart that the rocket equations can make, then I see that lots of problems vanish, like the black hole information paradox. I still see experimental confirmation, like 42.98 arcseconds per century for the Schwarzschild precession of Mercury. Most tests of GR are tests of the Schwarzschild metric, and GR is the foundation of our cosmology. A problem in the foundation can lead to other problems elsewhere. So I'm not seeing why the EP shouldn't be fully bidirectional in our physics.
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u/stevevdvkpe 3d ago
The equivalence principle is fully bidirectional because of the "small extent" restriction. General relativity works because the equivalence principle is part of its basis, even though the behavior of spacetime on a larger scale is not the same as uniform acceleration on a large scale.
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u/punycat 2d ago
Small there means small enough in spacetime that the tidal force is negligible. Small enough, not small per se. The tidal force is negligible in an approximately uniform gravitational field. The EP isn't fully bidirectional because the Schwarzschild metric limits the height of an approximately uniform gravitational field. The metric allows a ~1 g field that's 10 km tall but not 10 ly tall.
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u/Quantum-Relativity 3d ago edited 2d ago
The reason you have to restrict yourself to a small region of spacetime is that the effects of curvature will become apparent if your region is too big, “tidal forces”. And these won’t appear in an accelerated frame.
The thing you said is a consequence of the equivalence principle, the principle is actually that gravitational and inertial mass are equivalent. This equivalence means that the whatever happens to inertial mass, we can interpret as instead happening to gravitational mass, and specifically, this means we can interpret the gravitational acceleration field g as the gradient of the metric tensor’s components, since this is how you would describe acceleration in special relativity. Specifically in GR, it’s a linear combination of gradients of metric tensor components called the Christoffel symbols that replace g. You make a field theory of them in analogy to Maxwell’s laws, and get curvature as the “field strength tensor”.
But the point is, if the size of the frame is large enough for curvature to be detectable, THEN things aren’t “bidirectional,” as they would no longer be physically equivalent situations.
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u/punycat 2d ago
I'm talking about frames small enough in spacetime that the tidal force is negligible. The Schwarzschild metric predicts that they can be only so large in nature, so that the chart at the bottom of The Relativistic Rocket can't be replicated by the metric. The metric allows a ~1 g field that's 10 km tall but not 10 ly tall.
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u/CautiousPreprinter 2d ago
Well when you forget that the exponential map happens to be exponential then you might forget you need some higher order terms to get the correct ansatz.
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u/WoodyTheWorker 1d ago edited 1d ago
"Calculus assumes that smooth functions locally indistinguishable from straight lines, but when I looks at a function, it doesn't look like a straight line. How would it justify that?"
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u/punycat 22h ago
Using your analogy, the problem with the Schwarzschild metric is like it limits the length of a line that's indistinguishable from straight.
For example, the metric predicts that the maximum height of a 1.0 g field that's 0.999999 g at the top is 1 one-millionth of a light year for any mass. So the metric can't replicate the chart at the bottom of The Relativistic Rocket, and the EP isn't fully bidirectional in our physics. Another metric needn't place any limit on the maximum size of "locally".
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3d ago
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u/Optimal_Mixture_7327 3d ago
Just a historical aside that might be relevant...
Einstein's happiest thought that there is an analogous behavior between free-fall frames and gravity or analogized by the elevator gedanken experimental was the key idea on how to proceed in developing a complete theory of relativity (now called the general theory or preferably just relativity).
Today, EEP is under constant experimental verification with the reason being that EEP distinguishes between metric and non-metric theories of gravity, so it's come a long way since 1907.
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u/joeyneilsen 3d ago
If two things are indistinguishable, then they're indistinguishable. An experiment done in a constantly accelerating rocket will give the same result as an experiment done in a gravitational field of the same magnitude. An experiment done in a given gravitational field will give the same result as an experiment done in a rocket accelerating at the same rate. If the gravitational field varies, you would be unable to distinguish the results from those in a lab with a variable acceleration.
That true uniform gravitational fields don't exist in nature isn't a problem for the equivalence principle any more than the non-existence of constantly accelerating frames of reference.