12
votes
Is space — as opposed to space-time — curved by a gravitating mass?
Let's suppose you are an observer and you have a clock to measure time and rulers to measure distance. You construct a coordinate system by placing yourself at the origin and using your clock and ...
- 331k
9
votes
Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space
I think the core of your question is a very commonly-misunderstood subtlety, so I'll begin with a seemingly abstract example. Consider the vector space $V$ which consists of formal polynomials of ...
- 51.8k
6
votes
Accepted
Defining Surface gravity of a black hole
You're absolutely right. Because the Killing vectors are defined to satisfy Killing's equation
$$ 2\nabla_{(a} K_{b)} = 0 \ , $$
a constant scaling is possible, and the value of $\kappa$ changes. You'...
- 3,058
6
votes
Accepted
Is space — as opposed to space-time — curved by a gravitating mass?
A few facts:
In a 4-dimensional manifold such as spacetime you can pick any timelike direction and call it time in the vicinity of any given event. Directions orthogonal to this will then make up '...
- 47k
4
votes
Accepted
Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
Yes, the generalized coordinates $(q^1,\ldots, q^N)$ are assumed to be independent, i.e. no constraints, and the cotangent vectors $(\mathrm{d}q^1_p,\ldots,\mathrm{d}q^N_p)$ at each point $p$ are ...
- 170k
4
votes
Accepted
Is it necessary that the compactified manifold in string theory must be complex?
If we are talking about the traditional way of realizing the 10-dimensional target space of superstring theory as a product $$M^{10}~=~M^4 \times K^6,$$
where $(K^6,g^{(6)})$ is a compact $6$-...
- 170k
4
votes
How to understand physical tensors?
tldr.
For Q1, there is no one correct way of writing things. Describing inertia as a $(0,2)$ tensor as in the link, and as an endomorphism are both valid approaches.
Next, the term "a rank 2 ...
- 2,824
3
votes
How to quantify the idea that physical calculations of objects of close by geometry give same answer?
Often when one makes such approximations, they're usually shown to be valid under some set of assumptions, and the resulting calculations are usually 'continuous with respect to errors in the ...
- 2,824
3
votes
Accepted
Does a smooth world sheet imply strings moving faster than light?
TL;DR: The string is not moving faster than light.
A 2D tangent plane $T_pN$ of a regular/generic point $p$ of the 2D string world sheet $N:=X(\Sigma) \subseteq M$ imbedded into a Lorentzian target ...
- 170k
3
votes
Accepted
How are coordinates chosen in general relativity?
It seems as if the source was saying exactly what General Relativity is not. As pointed out in the comments, General Relativity a is covariant (coordinate independent) theory. In other words, it is ...
- 3,058
2
votes
Characterizing compactness of the Alexandrov topology in a spacetime
Consider a 2D Minkowski spacetime and identify two constant time surfaces $S_T\equiv S_{-T}$ for some $T>0$. Remove the lines $t=0$, $x\geq 1$ and $t=0$, $x\leq -1$ from the obtained cylinder. The ...
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2
votes
What is the physical idea of isometry of a metric?
You do require at least two points on the manifold to get a proper understanding of isometry. Isometry is a mapping that should preserve the proper distance between these two points. Since we define ...
- 591
2
votes
Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space
We should distinguish between vector spaces and the manifolds upon which they are tangent. A vector space is an abstract space where addition between the elements of the vector space is defined, as ...
- 591
2
votes
Is space — as opposed to space-time — curved by a gravitating mass?
"Can we relatively freely rotate our 4 dimensional coordinate system for the universe's spacetime such that what was space before (time fixed, say at zero) is rotated "into" the time ...
2
votes
How to quantify the idea that physical calculations of objects of close by geometry give same answer?
A general principle that we often tell students is to model
the system as simple as possible to try to capture the physics of interest.
(that is, Can one possibly eliminate a complication that ...
- 8,677
1
vote
What are the meaning of geodesics?
Newton's first and second laws of motion respectively state momentum is unchanged without a net force and the rate of momentum is equal to net force. Every student naturally wonders what the point is ...
- 22k
1
vote
Accepted
What is going wrong in my calculation of metric tensor for cylindrical coordinates?
The error is with the $\phi\hat{\phi}$ term. The position vector in cylindrical/spherical coordinates is NOT $\vec{r}=r\hat{r}+\phi\hat{\phi}+z\hat{z}$! This is obvious once you write down the ...
- 2,824
1
vote
How to understand physical tensors?
They are not equivalent definitions because they express two different cases of what an object a rank-2 tensor is. First, let's assume you have a tensor $\mathbf{T}$ in a $3 \times 3$ matrix ...
- 591
1
vote
Accepted
Abelian flat connection maps to zero
Hint: Given an abelian flat connection $F=\mathrm{d}A=0$, from Poincare Lemma we know that there exists a locally$^1$ defined $0$-form gauge transformation $\alpha$, so that 1-form gauge potential $A=\...
- 170k
1
vote
Question about Wald's example of a "derivative operator"
An explicit example may help. Consider the manifold $\mathbb R^2$ and a vector field $V$ which, in a Cartesian coordinate system $(x,y)$, takes the form
$$V = -y\frac{\partial}{\partial x} + x\frac{\...
- 51.8k
1
vote
Question about Wald's example of a "derivative operator"
He doesn't get to just define the object in some coordinate system and call it a tensor. And he can't just declare that its components, by definition, transform however they need to, so that the ...
- 7,876
1
vote
Apparent speed of an object in a circular orbit in Schwarzschild geometry
Yes, an orbiting object is the prototypical clock. Like any clock it is subject to gravitational time dilation.
- 65.7k
1
vote
Splitting of general pseudo-Riemannian Manifolds
There is a proof that we can always 3+1 foliate on a finite patch of a (smooth) spacetime. How large that patch is depends on the spacetime, of course, but if a caustic shows up, the expectation (I ...
- 38.8k
1
vote
Accepted
Divergence theorem in index notation
Maybe it helps you visualize if we expand things out. Let $A_{il} = \epsilon_{ijk}r_{j}\sigma_{kl}$, such that the first integral is:
$$\tau_{i} = \displaystyle{\int A_{il}n_{l} dA}$$
Notice that we ...
- 204
1
vote
How are coordinates chosen in general relativity?
The usual reason to choose a coordinate system in general relativity is convenience. For "convenience" you can read ease of calculation, ease of understanding, or similar ideas.
There is a ...
- 61
1
vote
How are coordinates chosen in general relativity?
Coordinate system is just a frame of reference in which you are studying the motion and relativity over it . And you are free to choose the frame of reference and so the coordinate system . For ...
1
vote
Mathematical characterisation of diffeomorphisms in General Relativity
GR is diffeomorphism invariant. If the spacetime manifold $M$ has a metric $g$, then it is possible that its diffeomorphism group $\mathrm{Diff}(M)$ has a subgroup which preserves its metric. This is ...
1
vote
What is the physical idea of isometry of a metric?
An isometry is a way of moving the manifold rigidly within itself, and is inherently an active transformation (a diffeomorphism).
From Carroll's Lecture Notes on General Relativity:
diffeomorphisms ...
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