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22 votes

I don’t understand Noether’s theorem… there is nothing to prove?

Comments to the post (v3): Noether's theorem is just one method to determine conservation laws. If you have another, that's totally fine. Not all quasi-symmetry transformations (which in principle ...
Qmechanic's user avatar
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10 votes
Accepted

Length contraction of a square moving along the diagonal

The mistake you made was assuming that length contraction is a linear operation. By linear here I mean in the mathematical sense. Length contraction is an operation that takes a position (or length) ...
Luke Pritchett's user avatar
9 votes

I don’t understand Noether’s theorem… there is nothing to prove?

Invariant Lagrangian and Conservation Laws: You're correct that if the Lagrangian is invariant under a transformation, it leads to a conserved quantity according to Noether's theorem. However, the ...
DimensionDestroyer's user avatar
9 votes
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Is it possible to describe every possible spacetime in Cartesian coordinates?

As the choice of coordinates is arbitrary, can't I just "postulate" to use cartesian coordinates to describe any possible spacetime? If by cartesian coordinates you mean a set of four ...
S.G's user avatar
  • 2,550
9 votes

Length contraction of a square moving along the diagonal

My line of thought was that we can resolve v into vx and vy, which have equal magnitude. Because of that, the horizontal and the vertical contractions will be equal and we will get a smaller square. ...
Dale's user avatar
  • 103k
9 votes

Will a distant observer really see an object that has fallen close to a black hole freeze in time?

One reason the Schwarzschild metric is so well known, is because it was the first exact solution found for a gravitational body in General Relativity and even Einstein did not think an exact solution ...
KDP's user avatar
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8 votes
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Mysterious Coordinates for Metric on a Sphere

As emir sezik said in a comment, this is the metric you get if you start with the Euclidean metric in $d+1$ dimensions with coordinates $(y^1, \ldots, y^d, z)$ and substitute $z = \sqrt{1 - \mathbf y^...
benrg's user avatar
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8 votes
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The synchronized clocks on earth's surface: at which observer's rate are they beating?

I assume this synchronized time rate is the proper-time rate of some observer.  Yes and no. ;) UTC is derived from TAI, International Atomic Time a high-precision atomic coordinate time standard ...
PM 2Ring's user avatar
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8 votes
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Generalized vs curvilinear coordinates

If you want to get to the heart of the matter, you need to understand the notion of a (smooth) manifold, whose definition will require you to understand what a (local) coordinate system/chart is. ‘...
peek-a-boo's user avatar
  • 6,685
8 votes

Can we define time as a field?

A field is just a function that is defined at each event in spacetime. So we could define a field $$f(t,x,y,z)=t$$ This would be time as a field. we always imagine time as if a forward moving field ...
Dale's user avatar
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8 votes
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Scalar multiple of inertial frame

You are mixing frames and coordinate vectors, and that is leading to some confusion. In a frame, vector quantities like position and velocity have some vector value. Changing frames may change those ...
Cort Ammon's user avatar
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7 votes
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The value of speed of light in different regions of spacetime

It is not a change in curvature that precipitates a change in the coefficients of the metric, but a change of coordinates. Indeed, at any event in any spacetime (regardless of how strongly curved it ...
Dale's user avatar
  • 103k
7 votes

Is it possible to describe every possible spacetime in Cartesian coordinates?

If OP by Cartesian coordinates means a local coordinate system $(x^0,x^1,x^2,x^3)$ [say, in some local open neighborhood $U\subseteq M$ of spacetime] such that the components $g_{\mu\nu}$ of the ...
Qmechanic's user avatar
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7 votes

Resolving an apparent contradiction between Schwarzschild and ingoing Eddington-Finkelstein coordinates

It is not a contradiction at all, and goes back to the well-known and often-emphasized fact that partial derivatives depend on which coordinates you keep constant. In general, if you have a function $...
peek-a-boo's user avatar
  • 6,685
7 votes
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Can we define time as a field?

A field is a quantity which typically depends on the location like --- giving a simple example --- a temperature distribution over a piece of metal, on one end of the metal the temperature is low and ...
Frederic Thomas's user avatar
7 votes
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Intuitive explanation of COSMIC TIME?

The cosmic time is defined as the (proper) time experienced by an observer at rest in a co-mobile frame, that is a frame carried by the expansion in an isotropic and homogeneous Universe (As such, it ...
Léo Vacher's user avatar
  • 1,143
7 votes

Do the Lorentz transformations lead to negative amounts of time?

The word 'event' here is a misnomer, because it carries with it the connotation of time, and the two events are simultaneous in the unprimed frame The word event is correct, not a misnomer. The fact ...
Dale's user avatar
  • 103k
7 votes

Why do we use the Earth inertial frame when solving problems involving non-inertial reference frames?

Why do we use the earth inertial frame when solving problems involving non-inertial reference frames? Because we can. This is the meaning of the general principle of relativity. You can use any ...
Dale's user avatar
  • 103k
7 votes
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Length Contraction: is $t'$ or $t = 0$?

David Morin is correct. Length in a given frame is defined as the distance between the two ends of an object at the same time in that given frame. So length is a concept that is intrinsically tied to ...
Dale's user avatar
  • 103k
6 votes

Derivation of the Schwarzschild metric: why are $g_{22}$ and $g_{33}$ the same as for flat spacetime?

I simply do not understand why, there, $g_{22}$ and $g_{33}$ must be those of the flat spacetime. In fact, it is not necessary that $g_{22}$ and $g_{33}$ take on the form of the angular part of the ...
Andrew's user avatar
  • 50.9k
6 votes

Is it possible to describe every possible spacetime in Cartesian coordinates?

Since the question makes no reference to the number of dimensions, you could ask it just as well for a universe that is 2-dimensions of space and 1 of time. If you can't do it even there, then the ...
NinjaDarth's user avatar
  • 2,093
6 votes
Accepted

Angular or tangential term in gravitational potential energy change using polar coordinates

The position vector in polar coordinates is $\vec r = r_r\hat r$ in your notation, or simply $\vec r = r\hat r$ without the redundant subscript. There is no component along $\hat \theta$. Consequently ...
Puk's user avatar
  • 13.8k
6 votes

Will a distant observer really see an object that has fallen close to a black hole freeze in time?

Tristan Diotte wrote: "if we switched to the "Eddington-Finkelstein" coordinate, this singularity would disapear" That doesn't help the far away observer, since the time ...
Yukterez's user avatar
  • 12.4k
6 votes

Scalar multiple of inertial frame

Yes. This is still an inertial frame. The transform you have posted is a change of units for the spatial coordinates. The Wikipedia list is not exhaustive.
Dale's user avatar
  • 103k
6 votes
Accepted

Change of variables from FRW metric to Newtonian gauge

Firstly as you suggested, $$ d t_{c} = \frac{\partial t_{c}}{\partial t} d t+\frac{\partial t_{c}}{\partial \vec{x}} d \vec{x}$$ $$d t_{c} = d t-\frac{1}{2} \dot{H}(t) \vec{x}^{2} d t-H(t) \vec{x} \,d ...
S.G's user avatar
  • 2,550
6 votes

Physical Quantities Sign Convention

Other answers have answered the direct questions in the OP. Here, I'd like to expand on the general question a little bit and talk about the fact that negative signs mean different things in ...
march's user avatar
  • 8,189
5 votes

Physical meaning of each component of the metric tensor in GR

This is like finding the meaning of individual coordinates of a classical position vector. The vertical component has something of a special meaning because gravity is vertical and it matters to just ...
mmesser314's user avatar
  • 40.7k
5 votes

Teacher told us we're not allowed to write negative vectors, is this correct or not?

Your teacher is instructing you to define the vectors in terms of their components, \begin{align} \vec F_1 &= \hat x \cdot 20\,\mathrm N \\ \vec F_2 &= \hat x \cdot (-10\,\mathrm N) \end{align}...
rob's user avatar
  • 91.4k
5 votes

Connection between pseudometric and Einstein elevator

lalala asked: I do not see how the metric tensor relates to accelerated reference frames. The purpose of the elevator experiment is to showcase that a uniform gravitational field can be switched off ...
S.G's user avatar
  • 2,550
5 votes

Resolving an apparent contradiction between Schwarzschild and ingoing Eddington-Finkelstein coordinates

The issue is that in general coordinate induced vector fields depend on all of the coordinates. For example, suppose we have coordinates $x, y, z$, and we change to $u, v, w$. Then even if the ...
Gleeson's user avatar
  • 213

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