# Tag Info

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No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work. You're thinking of something called "extrinsic curvature". In fact, many examples of extrinsic curvature - including ...

8

The answer to (2) is simply that no-one knows, and further that it's unlikely we will ever know. It's impossible to prove that the universe is infinite, but it's just possible we might prove it closed and therefore finite if the length scale is around the size of the currently observable universe. The paper Topology of the Universe: Theory and Observations ...

7

Two general methods come to mind: Prove that the Riemann tensor takes the form of equation 3.191, i.e. $$R_{abcd} = \frac{R}{d(d-1)}(g_{ac}g_{bd}-g_{ad}g_{bc})$$ If you are handed a metric, this should in principle be a straightforward calculation. If the metric is actually maximally symmetric, the calculation of the Riemann tensor usually turns out to ...

7

http://en.wikipedia.org/wiki/Light_cone It simply says that some parts of the space-time are not accessible to us. For example I assume :-) you are on (Earth, Now). No matter what you do (Moon, Now) is not accessible to you. (Moon, Now + 1 second) is also not accessible to you, because the Moon is 1.28 light seconds away from Earth. Some events from the ...

6

Yes, that's a fair description of what happens though of course from the ball's perspective it isn't moving - the rest of the universe is moving around it. However statements like this, while true, give little feel for what's going on. Actually it's extraordinarily difficult to get an intuitive feel for the way spacetime curvature works (or at least I find ...

4

For $p=1$, CTC's do not exist in Minkowski spacetime. In other $1+3$ spacetimes, in principle they are admitted in the absence of further requirements (like globally hyperbolicity) on the causal structure of the spacetime. They must be present if the spacetime is compact, for instance. For $p\geq 2$, the answer is obviously YES. Consider a manifold $M$ with ...

4

To me, the best way of describing it is as follows (semi-stolen from Carroll's Spacetime & Geometry): we all know from high school physics that, when no force acts on an object, it should not change its velocity $v$. In other words, the velocity vector tangent to the objects trajectory through space and time (i.e. world line) undergoes parallel ...

3

Nope, spacetime curvature says nothing about the dimensionality. Your intuition here is probably wrong because human imagination needs 'some dimension to bend into' in order for something to be curved (i.e. an embedding in a higher-dimensional space). This is just our lack of imagination showing, though.

2

You are right that there are two effects at play here. Firstly, suppose that we turn relativity off --- suppose that we consider our universe to be Newtonian, with a finite speed of light propagation. It would indeed be the case that if an observer A were moving relative to another observer B, and emitting a light signal towards B, then the rate at which ...

2

The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance. To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just: $$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2$$ Suppose you want to ...

2

I'm not an expert on this particular topic, but I believe I can answer your question. There are different kinds of "dimensions". The standard 3 spatial dimensions we live in are infinite in extent. However, one can also imagine dimensions that have a periodicity (like a circle). In such cases, there is a "size" to the dimension that refers to the ...

2

I'll raise some issues. Firstly you say In reality... Do you contrast something against something else here? It implies you say there the preceding sentence If you throw a ball, it will move along a parabola. Initially its vertical speed will be high, then it will slow down, and then speed up again... wasn't right, but that sentence seems pretty ...

2

First, let's assume we are working in Minkowski space just for simplicity (altough this can be argued too in a curved space-time using curves that connect the events/space-time points). This space has an affine structure, you can think of it as vectors connecting pairs of points. This vectors are geometrical objects and we don't need to specify a basis for ...

1

As to the first paragraph, gravity shows up as geodesic deviation; initially parallel geodesics do not remain parallel. Since, for a freely falling particle, the proper acceleration (the reading of an accelerometer attached to the particle) is zero, it is correct to say that a particle whose worldline is a geodesic has no proper acceleration. But it is not ...

1

To really understand this you should study the differential geometry of geodesics in curved spacetimes. I'll try to provide a simplified explanation. Even objects "at rest" (in a given reference frame) are actually moving through spacetime, because spacetime is not just space, but also time: apple is "getting older" - moving through time. The "velocity" ...

1

Time is relative. When it comes to Time Dilation, you actually see dilated time of another observer. So, your own time flow won't get frozen in any case. Hypothetically, you can see another one's time frozen if she is traveling at speed of light (time dilation by speed) or she is at event horizon of Black Holes (gravitational time dilation). Unfortunately, ...

1

We model spacetime as a manifold and a metric. Broadly, the manifold gives us the dimensionality and connectivity while the metric provides a method of specifying distances. The equations of General Relativity allow us to calculate the metric from the stress-energy tensor (or vice versa if you're Miguel Alcubierre). The point that jinawee is making in his ...

1

The (nearby) "separation between objects" you are referring to is the space-time metric. A metric in cosmology describes the expansion of space on large angular scales (low $\ell$ on the angular power spectrum of the universe). Without going into the mathematics, the expansion of space is driven by cosmic inflation, and is affected by things the amount and ...

1

How about this for a more "physical" definition: two points in space-time are time-like separated if and only if a massive particle starting at one could, if subjected to appropriate finite forces, reach the other. Replace "massive" with "massless" to get the definition of light-like separation. If neither is possible the the points are space-like ...

1

Let $\lambda, \mu, \nu$ be functions on the reals to points (events) in spacetime. Let these be "straight" curves, in the sense that $\lambda', \mu', \nu'$ each all have the same direction for all values of their parameters. For example, $\lambda(u) = \lambda_0 + lu$ is a simple case, as $\lambda'(u) = l$. The vector $l$ is the vector along the direction ...

1

I am not sure wether I unstood your question correctly. From what I understood, you asked wheter particles are only connected/interacting by forces. Probably this is a matter of taste question but the picture of forces gets very unconvenient when one is talking about paulis exclusion principle. Although it can not be put in terms of a simple force it has a ...

1

Theoretical viewpoint: Einstein field equations can be written in the form: $$\color{blue}{G_{\mu\nu}}=\color{red}{\frac{8\pi G}{c^{4}}} \color{darkgreen}{T_{\mu\nu}}$$ We can write in simple terms: $$\rm \color{blue}{Space-time \,\,geometry}=\color{red}{const.}\,\,\color{darkgreen}{Material \,\,objects}.$$ And the $T_{\mu\nu}$ is a mathematical object (a ...

1

Is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then to B it APPEARS that A's clock moving slower, but A sees his own clock moving at normal speed. Similarly, to A it APPEARS that B's clock is moving slower, but B sees his own clock moving at normal speed. This is true. If the above is true, then ...

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