Spacetime Definition and 1000 Threads

Hi, Killing equation in flat space is just \partial_{\mu}K_{\nu}+\partial_{\nu}K_{\mu}=0 . I've seen in various places the solutions to this written as K^{\mu}=\Lambda^{\mu}_{\nu}x^{\nu}+P^{\mu} where P is a constant 4 vector, and \Lambda_{(\mu\nu)}=0 (i.e. symmetric part vanishes and it is...

In section 1.2 of Taylor and Wheeler's Spacetime Physics, a rocket moves past a laboratory (on Earth). Attached to the rocket is a pin. From that pin a spark is emitted at two locations in the lab, separated by 2 meters. The observer in the rocket measures the elapsed time between the sparks, as...

In figure 3-1 (page 63) of Taylor and Wheeler's Spacetime Physics, the observer on the train determines that the lightning strikes are not simultaneous because the flashes do not reach her simultaneously. I see two problems with this. 1. The narrative in figure 3-1 contradicts the text in...

In another thread, https://www.physicsforums.com/showpost.php?p=2973770&postcount=45, some questions came up about what the conditions are for a spacetime to admit flat spatial slices, and for a spacetime to have a time-independent "scale factor" (see definition below). These questions...

What is the difference between the Algebra of Physical Space (APS) and the Spacetime Algebra (STA), and why do we need them both?

General particle motion in an FRW spacetime is governed by the geodesic equation, \frac{du^\mu}{ds} + \Gamma^\mu{}_{\nu \alpha} u^\nu u^\alpha=0, with u^\mu = \frac{dx^\mu}{ds} and where the Christoffel symbols are given in the notes. Show that the physical momentum of any particle always falls...

Homework Statement A free particle is moving in the x direction through Minkowski spacetime, and has velocity V as measured by a stationary observer at x = 0; t = 0. Express the particle's world-line parametrically in terms of V , parametrized by the particle's proper time  Homework...

Do the Eddington-Finkelstein coordinates allow to cover the maximal analytic extension of the Schwarzschild spacetime? ans if not what region do they cover?

First post, I am just a layman, so go easy. The way I understand it, the moon is following a straight line through space. However, the mass of the Earth is so great that it warps the fabric of spacetime and it appears that the moon is going around the earth. Same with any moon or planet, they...

So I have just recently started thinking about the universe and things like that, and I had a question about spacetime. I was curious as to how exactly spacetime works in the universe. Pictures usually show only one plane of spacetime, such as this picture...

"Pauli matrices with two spacetime indices" Hi all. This is my first post so forgive me if my latex doesn't show up correctly. I am familiar with defining a zeroth Pauli matrix as the 2x2 identity matrix to construct a four-vector of 2x2 matrices, $\sigma^\mu$. I'm trying to read a paper...

In my reading I came across the equation ds2 = −dt2 + 2t/r dtdr + (1 − (t/r)2)dr2 + (BKr)2(dθ2 + sin2 θdϕ2) where s is spacetime, t is time, r is radius and the others are not important for my question. What I do not get is the "1" in the (1 − (t/r)2)dr2, or dr2− ((t/r)2)dr2 . This seems...

What is Nima talking about at 42 minutes? http://www.cornell.edu/video/?videoID=913&startSecs=0

If we start with the Lorentz transformation \begin{align*} ct' &= \gamma (ct - \beta x) \\ x' &= \gamma (x - \beta ct) \\ y' &= y \\ z' &= z \end{align*} with the usual \beta = v/c, \gamma = 1/\sqrt{1-\beta^2} and take the limit c \rightarrow \infty, then we get: \begin{align*} t' &= t \\...

From what I've read, String Theory is a theory of everything, unlike some of the other quantum gravity theories. That means that String Theory explains other particles and fields including gravitons. So String Theory is a quantum gravity theory because it includes gravitons. But I was...

General relativity has it that the spacetime continuum is curved. The physics of continuum is dealt with [stress] tensors. My questions: (1) The presence of a mass creates the curvature in spacetime. By how? (2) If the curvature due to matter is positive, is the curvature due to antimatter...

I was just gong to learn general relativity(not with maths) but with some very basic tutorials given over internet. I also watched the animated series of general realtivity. Everywhere i see,matter bends spacetime( a fabric of spac and time woven ). And when there is matter than this...

For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...

We have described the distortion in spacetime which Einstein derived in GR as a "curvature" of spacetime. This is barely more descriptive than "warping" spacetime. I understand that what this means is that spacetime varies from being Euclidean, having distortion caused around objects of mass...

hi, how does general relativity work INSIDE stars and planets, since the mass is no longer concentrated within a point, so there are necessarily gravitationnal effects outwards and not only inwards?

When an object moves in curved spacetime, how does its angular momentum vector transform? My hunch is that the curvature tensor is involved somehow, but can't figure out an obvious equation for dLa/dxb. Probably I even wrote that derivative wrong .. sorry I'm a newbie at GR.

Hello all I am trying to teach myself general relativity and am working through the text 'a first course in general relativity' by Bernard F Schutz. So far I have made slow but consistent progress but I am perplexed by his derivation of the ‘local flatness’ result. This says that for any point...

Hi all, I'm going to ask a naive question - hope that's ok. There's been a lot of recent discussion of the results from Webb et al. which indicate that the fine structure constant varies spatially. I realize the results are very controversial - I'm wondering, hypothetically, if these...

When people discuss the Schwinger model, sometimes they still call the electron field spin-1/2 and the EM field spin-1. I wonder if there's some justification for these calling, since there's no rotations at all in 1+1 spacetime. I know for SO(n) with n>=2, one can always have well-defined spins.

So, I have a simple question about these two. All matter can ultimately be broken down to energy, and then there's space-time. So are these two the only things that exist in our universe? Can either of them be broken down even further and if so, could they ultimately be interchangeable? And...

This is not a mathematically supported proof or a detailed one, but rather a philosophical proof, being my (temporary) conclusion of https://www.physicsforums.com/showthread.php?t=404650". To sum up the initial problem, there are two bodies in space rotating about one another, attracted by...

I know of the gravitational analogy. The bending of spacetime due to a mass is analogous to a ball placed on a sheet, other balls in the region will be "attracted" towards each other. My question is, if we have to simplify our 3 spatial dimensions to 2 dimensions for the analogy, does that...

I know that spacetime is a Lorentzian manifold, but what kind of properties has to be required exactly? for example orientable, connected, Haussdorff, ...

Special relativity should be a special case of general relativity, for flat spacetime manifolds. For locally flat manifolds, special relativity should however give approximate results. But even Earth is a non-inertial frame. So that would mean that special relativity can only be observed for...

Just wondering. The Hubble looks back 13 billion light years and photographs galaxies in their early formative stages not too long after the Big bang. Now let's say 13 billion years from now, if man and woman and galaxies are still alive, the version of the Hubble, in the year 13,000,002,010AD...

Homework Statement Paraphrase: A Transporter can reduce a person to data and transmits the data by light or radio signal to another location. A person is beamed from Earth to the planet Zircon orbiting a star in the Andromeda Nebula, two million light-years from Earth. Neglect any relative...

Can the strings of string theory live in only 4D spacetime? Do we get anything interesting or useful? What do we lose? Thanks for any ideas or help!

Interesting paper: http://arxiv.org/abs/1009.1136v1 The Small Scale Structure of Spacetime Steven Carlip (Submitted on 6 Sep 2010) Abstract: Several lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional. I summarize the evidence for such...

i recently had trouble understanding some concepts with one of my astronomy assignments. I understand that the universe is expanding, and that galaxies further away from each other are expanding at a faster rate than galaxies closer together, yet the space inside a galaxy remains constant due...

Simple Experiment. Find a bookshelf full of books in a well lit area (such as a library or bookstore.) Find a seat about 10 feet away facing the bookshelf. Hold out your right hand in a fist with your thumb pointing out little less than a foot from your face just outside your peripheral vision...

Hi, How can we represent covariant and contravariant vectors on curved spacetime diagrams? How can we draw these vectors on a spacetime diagram? Contravariant vectors are really vectors, therefore we can represent them on the diagram with directed line elements. Covariant vectors are...

I'm here to discuss motion in spacetime and how it works to hopefully get a better understanding of it. Specifically "spacetime swimming", and the motion of a "relativistic glider", which is talked about in this article "Surprises from General Relativity: "Swimming" in Spacetime" By Eduardo...

Maxwell's equations in curved spacetime can be written as \nabla^a F_{ab} = -4\pi j_b, \nabla_{[a} F_{bc] = 0 or as d*F = 4\pi*j, dF = 0, where F is a two-form, j is a one-form and * is the Hodge star. How do you show that these two sets of equations are equivalent (basically, that the first...

Background: Math: An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics. A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature. Given the existence of a connection with Riemann...

There are many derivations of E=mc2 out there, but did Einstein actually used Minkowski space time for his original derivation of E=mc2? How did he do it?

Does string theory merge space and time into spacetime? GR combines space and time as spacetime, I've heard that in string theory there is 9+1 or 10+1 spatial dimensions, with 3 large, 6 curled, 1 time dimension. Is there a spacetime in string theory? Are Yau-Calibi manifolds part of this...

Hey When someone says that spacetime is stretched or compressed, is it meant to be taken literally? If so how would one determine which areas of spacetime are thicker/thinner (i know we can look at the metric but I am talking about experimentally how would we know)? Or is this just another...

So, I don't get it... I know Einstein said that space-time was a "fabric" but I can't visualize that, with us being in three dimensions.i Can visualize in my head (or at least understand) how space and time are really the same thing (or to my knowledge, time is just a component of space, a...

I've got a simple question I can't find any quick answer to. I understand that if various observers with different relative kinetic energy (velocity) are to measure the speed of light of the same event the same (c), time and space values must be different for them. But how do we know which...

A question in Sean Carroll's spacetime and geometry, 4.3 (I have solved and removed the first question posted before, only the second left) 1. Homework Statement Hi, my questions is , how to derive eqn (4.64) \delta \Gamma_{\mu \nu}^{\sigma} = - \frac{1}{2} [ g_{\lambda \mu} \nabla_{\nu}...

The mass of an object warps/bends/disturbs spacetime. But does the active warping/bending of spacetime create mass/energy?

OK, i have just been watching a documentary about time (sorry if this is not allowed but it is here: http://www.youtube.com/profile?user=turxxx#p/c/B6BE0700688DBF9D/0/V3aYKAJEVfQ) anyway i think i now understand how time changes rate for a person (i am actually going to use time dilation and...

The backdrop against which cosmology is set is provided by the geometry of spacetime. As often pointed out in this forum, General Relativity has revealed that this geometry is not fixed and eternal, but changes as the structure of the universe evolves from an ultradense, ultrahot but formless...

I know that folding spacetime is completely possible, and I'm pretty sure that the answer I'm looking for is in Einstein's field equations. I just don't know enough about tensor calculus (yet) to figure this out for myself. So my question: how much energy is required to completely fold a certain...

As I understand it, anything that is traveling at the speed of light moves through space but not through time. So if we were able to track a photon from the surface of the Sun, we would say it took about 8 minutes for the photon to travel to Earth. However from the photon's point of view, it...