Spacetime Definition and 1000 Threads

In Minkowski spacetime, calculate ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}##. I had calculated previously that ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}## When I subsitute it back into the expression...

Firstly, since ##\{ \mathbb{I}, \sigma_x, \sigma_y, \sigma_z \}## is a basis of the space of ##2 \times 2## Hermitian matrices, and because ##X = t \mathbb{I} + x\sigma_x - y \sigma_y + z \sigma_z##, the map is one-to-one (because each matrix has unique decomposition). It's also easily checked...

Deriving time dilation was easy: Imagine two events in frame O' at the same location. ##ds^2 = -c^2 dt'^2## The same viewed in O frame is: ##ds^2 = dx^2+dy^2 + dz^2 - c^2 dt^2## ##\Rightarrow dx^2+dy^2 + dz^2 - c^2 dt^2 = -c^2 dt'^2## ##\Rightarrow (\frac{dx}{dt})^2+(\frac{dy}{dt})^2+...

Summary:: Sci-Fi author looking for science advisor Hi everyone :) I have just completed the first draft of a novel and am looking for someone to review the science and confirm I'm not wildly off base, misunderstanding, or otherwise talking out of my ass before I begin the edits for the...

https://scipost.org/SciPostPhysLectNotes.10/pdf "Space expands behind the warp bubble and contracts in front of it, thus pushing the bubble forward at velocity v. The ship, which is at rest inside the bubble, moves along with the bubble at an arbitrarily large global velocity." If the ship...

Hi guys, I'll attach an excerpt from my textbook which isn't, in my opinion, very clear in explaining a spacetime interval(or I'm just missing the key to get the concept). "How do we combine two different measurements such as time and space, to form an invariant variable? We can simply write...

Hello, Some doubt arose me reading this thread https://www.physicsforums.com/threads/is-acceleration-absolute-or-relative-revisited.999420/post-6454462 currently closed. Sorry, I have not be able to quote directly from it :frown: Your claim is not , however, asserting that the spacetime...

I have one question I hope someone here can answer for me. Relativity theory tells us that space and time are sort of the same thing, as a spacetime. So when space is expanding, what happens to time? I find it hard to believe that time is somehow unaffected by the expansion of space, so while...

First, in Anastopoulos C, Hu B L. A master equation for gravitational decoherence: probing the textures of spacetime[J]. Classical and Quantum Gravity, 2013, 30(16): 165007. , the Einstein-Hilbert action is used to analysis a quantum matter field interacting with the gravitational field...

Hello, I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR. All us know the physical meaning of timelike paths: basically they are paths followed through...

Spacetime diagrams seem to be the most used explanation for relativity weirdness, so I’d like some clarification in how to make them, it anyone wants to help. (1) Light’s worldline is 45 degrees, obviously. No issues there, I don’t think. (2) How do I determine the angles of the moving frame...

[Mentors' note: This thead was forked from another thread - hence the reference to "these replies" in the first post] I am wondering why all these replies only discuss Lorentz transformations in 1+1 spacetime dimensions. That is the easy bit. The problems in understanding arise in 2+1...

Hello, here on PF I've seen many threads about the concepts of 'reference frame' and 'coordinate system'. In the context of SR my 'envision' about the concept of 'frame of reference' is basically the 'rods & clocks latticework' as introduced in the book Spacetime physics (Taylor, Wheeler)...

Tad Williams’ Otherland series has a scene where the characters are drawn to a temple no matter which direction they try to walk, as if space itself is curved. This is kind of the intuition I get when a physicist talks about the spacetime interval kind of flipping past an event horizon: if you...

Hi everybody I saw quite a nice Youtube vid about general relativity and how gravity bends spacetime and therefor redirects angular momentum into the center of gravity. I thought the first time I begun to understand the concept but immediatly the questions poped up. The video basically says...

Hello, i can't understand how does the author found this expression relating ##x_{c}## and v. I already tried by a lot of geometrical ways, knowing that the tangent of the angle between the dotted line and the x-axis should be v, but the results are illogical. Could you help me? I am start to...

[Moderator's note: Thread spun off to allow discussion of this topic to continue since the previous thread was closed.] I have had something nagging at me about this for a while, and it finally hit me while looking through this paper about the Godel Universe...

The distance/difference between two points in spacetime can be written in two forms (as shown in attachment). Can anyone explain the difference in the two equations? I have read that the two equations are the same, but i don't understand the change in sign. Why is it written in two forms...

Okay so for this one we can consider the Bianchi identity again$$\begin{align*} \nabla_{[\lambda}R_{\sigma \rho] \mu \nu} = 2 \nabla_{\lambda} R_{\sigma \rho \mu \nu} + 2\nabla_{\rho} R_{\lambda \sigma \mu \nu} + 2\nabla_{\sigma} R_{\rho \lambda \mu \nu} &= 0 \\ \nabla^{\lambda}...

I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##. So we have: $$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$ Now I know that ##dx'^{\mu}=C_\beta \ ^\mu dx^\beta## and ##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma## where...

I enjoy explaining spacetime curvature to people with a rank-beginner understanding of GR. But someone asked about that favorite concept in pop-sci, spaghettification. I'm having a hard time with it. If you fell into a black hole, there's no reference frame within which you could describe...

For the flat spacetime we could just use that partial derivatives commute as well as the antisymmetry of ##F^{ab}##, i.e. ##\partial_b \partial_a F^{ab} = -\partial_b \partial_a F^{ba} = -\partial_a \partial_b F^{ba} = -\partial_b \partial_a F^{ab} \implies \partial_b \partial_a F^{ab} = - 4\pi...

I just learned from the American Journal of Physics that the two books Space Time Physics by Taylor and Wheeler and Exploring Black Holes by Tayor, Wheeler, and Bertschinger are for free now! What a nice Christmas gift! http://www.eftaylor.com/spacetimephysics/...

In Phillip Harris' (U. Sussex) post on special relativity he includes on p. 45 an algebraic proof of invariance of spacetime intervals. He starts with the definition S^2 =c^t^2 - x^2 -y^2 -z^2, he inserts the Lorentz transform expressions fot t and x, and he does some algebra to show that one...

Via web search found https://www.physicsforums.com/threads/what-dimension-does-space-time-curve-in.852103/ Read it and watched two videos mentioned: I understand we cannot perceive 5D ;-), so extrinsic visualization of maximum of 2D intrinsic curvature is possible. So time+1d space is all we...

Hello there.We know that spacetime may have singularities and the current theories can not describe it very much.I want to start reading about quantum gravity but what is the progress done so far for the resolution of questions about the singularity?Could a different approach perhaps a...

Hello there.The question is as stated:does light curve spacetime?We know that bodies with mass do curve spacetime but does a massless particle or wave like light curve spacetime?Thank you.

It seems a gravitational field does not alter the electromagnetic field strength. Is this correct? My reasoning: With no gravity, field strength is: F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu Introduce gravity: \partial_\mu A_\nu \rightarrow \nabla_\mu A_\nu = \partial_\mu A_\nu +...

Ispired by PeterDonis remark about "river model" in some thread a time ago I made next visualization picture. The graph desctibes, how the flat Minkowski spacetime is changed in presence of mass (black hole). It do not need much explanation, almost everything is described at the picture. To me...

I'm a bit confused about GR : what is more significant about the considered spacetime, the metric, which is time-independent, or the embedding (there are already some posts on PF about it), which describes the shape of a manifold, but is time-dependent ?

I want to know whether Quantum Fluctuations could exist without the presence of Spacetime. Would it be possible, in the event of a Big Rip scenario, and if Spacetime really would get ripped apart, that quantum fluctuations could still occur? And if Spacetime is ripped apart, does that mean the...

Hi, My question can result a bit odd. Consider flat spacetime. We know that inertial motions are defined by 'zero proper acceleration'. Suppose there exist just one free body in the context of SR flat spacetime (an accelerometer attached to it reads zero). We know that 'zero proper...

As far as I know, the grand prize of a Theory Of Everything is mathematically uniting of all forces in the conditions close to the big bang but one of the main problems from the GR end of things is that gravity is not actually a real force to be combined with anything. All the most popular...

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Hey there, I'm aware this is a bit of a stupid question, and I think that I understand the principle fundamentally, however, my intuition is still having a little trouble catching up, and I'm trying to figure out if it is because of an important detail that I have missed/misinterpreted. I think...

My question in specific is understanding what this line EB exactly represents. This was borrowed from the book "A first course in general relativity" by Schutz. There is a question on page 30 (number 12) which asks the following: "Use the fact that the tangent to the hyperbola DB in Fig. 1.14 is...

I had a thought that I wanted to share in another thread, but it wandered way off track and quite properly was closed. But I thought the separate idea that I had spawned from the old thread was worthy of posting in a new thread. I do not want to re-open the old thread, though! In flat...

1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities $$Q^t :=...

When people try to explain how gravity works, the following example is constantly used . However, I don’t understand how this explains HOW gravity works. By using this example, gravity itself is used as a bias to explain how gravity works. How can explain gravity by saying “things fall along...

Hi, in general relativity I'm aware of the spacetime 'distance' between two timelike related events is maximized by the free falling timelike path (zero proper acceleration) joining them. Consider now a couple of events belonging to a spacelike hypersurface (AFAIK it is an hypersurface with...

What is Gravitational dragging ?

I'm studying differential geometry basics for general relativity (no specific source, just googling around). I know that spacetime is modeled as a ##4##-dimensional smooth manifold. Smooth manifold means that we consider a restriction of the maximal atlas such that all charts in it are...

I was just reading about de Sitter space and the following question occurred to me: de Sitter spacetime is curved despite containing no mass-energy, because it has a positive cosmological constant. Does it have a foliation into spatially flat, constant-time hypersurfaces though? Maybe it's...

This approach is seeming intuitive to me as I can visualize what's going on at each step and there's not much complex math. But I'm not sure if I'm on the right track or if I'm making some mistakes. Here it is: ##A## has set up a space-time co-ordinate system with some arbitrary event along his...

I was looking at this chart and I didn't understand how increased angular momentum of the test particle curves the spacetime around the center mass. If that is how it's interpreted. Now the way it looks like is that the curvature is dependent on the angular momentum of the test particle.

In de Sitter-Schwarzschild spacetime things close to the black hole are falling towards it whereas in greater distance they are receding. So there should be a certain (unstable) ##r##-coordinate, where things are static. The de Sitter-Schwarzschild metric has according to Wikipedia...

For someone who have just finished the study of the (fundamentals) of Schwarzschild spacetime, what would be the next natural topic to study?

In a spacetime diagram the spatialized time direction is the vertical y-axis and the pure space direction is the horizontal x-axis, ct and x, respectively. The faster you go and therefore the more kinetic energy you have, you'll have a greater component of your spacetime vector in the...

Since it is nonlinear, the 3 leg lengths would be limited to differentials? But how would the metric coefficients be incorporated into those leg lengths? It seems like the leg differential lengths would have to vary inversely with the magnitudes of the metric coefficients? For example, near...

If I make two rods with 1 meter length here on the surface of earth, and send one of them near a black hole that is at rest relative to earth, placing it there with its length alligned in the radial direction of the black hole, would I see the rod close to the black hole with a length shorter...