I apologize if this has been asked previously or if my thinking is way off base, but I am inexperienced with relativity (and this is my first question on the site).

I am wondering; as one (not literally - I am ignoring the death and pasta-related calamity, I'm basically just thinking in terms of theory and math, not the tidal contortion of human bodies) approaches the Schwarzschild radius of a black hole (or approaches the speed of light - I think that these are related in terms of relativistic transformations, but please correct me if I am wrong), Lorentz transformation dictate the length is contracted and time is dilated. At the actual event horizon, spacetime intervals are null.

Then beyond $R_{s}$ or $c$, space-like intervals become time-like and vice versa: the interval $ds^2\, =\,-c^2dt^2\, +\, dx^2\, +dy^2\, +\, dz^2$ that describes our familiar, day-to-day spacelike intervals, falls to zero and then becomes negative, marking the reorientation of the light cone. Space becomes timelike (forward march toward singularity) and time becomes spacelike (um.).

So what does this mean for wave properties such as frequency or wavelength? Does the frequency or period of an electromagnetic wave actually (or maybe just mathematically or theoretically) become a measure of distance? Does wavelength become like a measure of time?

I might be taking this reversal of interval thing too literally, but I'm genuinely curious, and thank you for any responses or corrections.


Yes, you are taking the reversal thing too seriously. If you used Kruskal-Szekeres coordinates then those coordinates don't flip from timelike to spacelike. The flip is just because you chose bad coordinates. If you had flat boring Minkowski spacetime you could pick a coordinate system where a coordinate flips from spacelike to timelike across some surface. It does not mean anything at all happened at that surface.

As for wavelengths and frequencies: wavelength and frequency are completely frame dependent quantities, the same electromagnetic field will have absolutely every possible wavelength to different observers, and will have absolutely every possible frequency to different observers.

Now let's talk about time dilation and length contraction. These are ideas from Special Relativity about about how different frames disagree about simultaneity. You judge time between two events as which surface of simultaneity intersect the two events. So different observers have different surfaces of simultaneity and so assign a different time interval to the same two events. But the proper time along a curve is something everyone agrees on, and it's what a clock traveling between those events actually records. And yes, it depends on the exact path and speed. Length contraction is about the length within a surface of simultaneity. So different surfaces actually consider different events, so you are talking about distances between different events.

So why do I bring that up? Because there isn't a unique surface of simultaneity for a local observer in General Relativity, so time dilation can, at best, can be about two different observers comparing with each other.

The real physics is that clocks measure the proper length along their world lines as given by the metric at the events through which the clocks travels. It's actually that simple.

You have events. You have a metric. You have curves that go through some events. The curves have an arc length determined by the metric at those events. When you have a second of arc length from a tick at event A, you get another tick. When you have two seconds of arc length from a tick at event A you get two ticks.

So you get curves and you know where along the curves the clock ticks. And that is it. It isn't mysterious. You just have to accept that clocks tick based on the metric it experiences along the path it takes. Because that's what clocks do. They don't do something else. They never did do anything else.

So as you approach an event horizon you still take a path through spacetime and the metric along that path determines which events have a tick of the clock. That's it.


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