This equation {\displaystyle T_{d}(h)=\exp \left[{\frac {1}{c^{2}}}\int _{0}^{h}g(h')dh'\right]} is the one that the OP used, I think. You can put any g(h) in that integral - and if you put a g that is constant, you still get a non-zero time dilation.
This article from physics stack exchange general relativity - Does gravitational time dilation happen due to height or difference in the strength of the field? - Physics Stack Exchange might be helpful, as might this The Relativistic Rocket
Ah, still makes me head esplode a bit. I can follow the equation, but that still didn’t say why.
But when they started talking about red shift, that made it make more sense.
Of course the light coming up is going to be red shifted, and the light going down is going to be blue shifted. The observer at the top will observe the lower clock ticking away at a slower rate than his own, and vice versa. In Newtonian mechanics, it would take longer for the light from the bottom to get to the top than vice versa, and since that violates relativity in that it should take light the same time to traverse the distance either way, then time must be what gives.
Once I thought about it that way, it made sense.
I think that made sense, anyway.
TIL, thanks.
By the way, I think Einstein’s derivation of gravitational time dilation explains it by first showing time dilation for the elevator/rocket case and then using the equivalence principle to argue that the same would apply for the gravity case.
The syntax is really easy to learn. The first equation starts out:
t_0=\sqrt{1-{2GM\over rc^2}}
How hard is that? _ gives subscripts, \sqrt{…} builds a big square root sign over what’s inside the braces 1- is clear and \over (admittedly non-standard LaTeX, but it works) gives a fraction of what is between that pair of braces. The whole surrounded in dollar signs to put it in math mode. You could learn enough in an hour to do that sort of simple equation.
The formula itself I got from Wikipedia, so blame any quirks on them.
I did develop an okayish flow for posting LaTeX here. It’s more cumbersome than it should be, but not too bad.
\phantom{.png. Note that this is invalid LaTeX, but that’s ok and is required for Discourse to pick it up.https://latex.codecogs.com/png.latex?. This is a site that will take the input LaTeX and render it to an image.See this post for several examples of the technique. Once you get going, and have a browser tabset going with the right tools, it only takes an extra 10 seconds or so to convert the LaTeX into a pasteable URL.
It clearly comes as no surprise to you that at work, I am in fact the local “regex guy.”
Thanks both of you
I did spend some time w LaTeX tutorials and it’s real straightforward. The url encoding is the gunky part to do by hand. But armed with yet another online calculator to do it for you, that too is trivial.
Not sure when I’ll next be posting algebra, but when I do, look out World!
Duly warned!
I think the problem with finding a comparable solution in general relativty for an inifnite plane in Newtonian gravity is not that it will collapse*, more that what is the most comparable as an infinite plane, unlike the Newtonian case, you have to consider the background and how it relates to the infite plane. Analogous solutions do exist though.
*In asymptocally flat spacetime there will be a limit to the extent that you can build a wall (depending on thickness, density, etc), but a spacetime containing an infinite wall can’t asymptoically flat. Cosmological solutions don’t necessarily collapse for the same reasons.
Thanks.
What I like about the constant gravity examples for gravitational time dilation is that it separates out any independent effect due to differences in gravitational acceleration on the clock. A pendulum clock reads differently at the top of a tower than at the bottom due both to GTD and to the difference in gravitational acceleration, but the rocket example (or infinite plane example) eliminate the difference in gravity effect (as would using a spring clock, of course).
An apparently different way of looking at it is that gravity is caused by the time dilation.
Gonna be honest, I more or less followed what he was saying, but haven’t grokked it well enough to rephrase in my own words.
I assume that he’s not just talking out his ass, anyway.
There is a congruence between Schwarzschild observers in Schwarzschild spacetime and Rindler observers in Minkowski spacetime. What this means in simpler terms is that static observers arranged radially in a speherically symmetric gravitational filed get the same results from comparing their clocks as observers arranged along the length of a (Born-rigid) accelerating rocket ship. This isn’t quite the same as the equivalence principle though.
I suppose that you can frame it that way, but all you can actually show is that the mathematical relationship exists: You can’t really say which one “causes” the other.
I don’t think that video is that great. What he does say is too dumbed down IMO to give real insight - gravity in GR is modelled as spacetime curvature, not as time dilation or “time curvature” or space curvature for that matter.
Just to expand on why I don’t like the video: saying gravity is tiime dilation can be made to work when dealing with static (time-independent) gravitational fields using the weak-field approximation for GR (in which gravity can be described as a potential). In this regime you can derive a gravitational time dilation factor from the potential and vice versa. I.e. you can describe a static weak gravitational field, purely in terms of time dilation.
But gravitational fields are generally not static and the weak-field approximation, as the name suggests, is just an approximation of GR that works only for weak fields. In GR as whole, there is no general procedure for comparing the clocks of different observers, so graviational time dilation, despite being an important physical prediction, is not a fundamental concept of GR.
Fair enough. I don’t usually watch his videos, as they are usually a bit dumbed down, and I don’t love his presentation style, but that one happened to come up in my feed in the middle of this discussion.
Good to know. I’d watched some of his videos and wasn’t entirely sure how I felt about some of the material.
SMH.
Okay, Both of you have A’s, and both are just about the same color. Now that we’ve gone to this board, I have been ID’ing people by avatar more than name.
I just realized that @Andy_L got swapped out for @Asympotically_fat.
Tag-team!