All Questions
Tagged with metric-field or metric-tensor
3,672 questions
110
votes
7
answers
134k
views
What do spacelike, timelike and lightlike spacetime interval really mean?
Suppose we have two events $(x_1,y_1,z_1,t_1)$ and $(x_2,y_2,z_2,t_2)$. Then we can define
$$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2,$$
which is called the spacetime ...
- 37.4k
107
votes
4
answers
11k
views
Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
- 1,357
104
votes
2
answers
8k
views
Is there such thing as imaginary time dilation?
When I was doing research on General Relativity, I found Einstein's equation for Gravitational Time Dilation. I discovered that when you plugged in a large enough value for $M$ (around $10^{19}$ ...
- 4,017
73
votes
2
answers
9k
views
Is spacetime flat inside a spherical shell?
In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the ...
- 2,193
71
votes
5
answers
24k
views
Conformal transformation/ Weyl scaling are they two different things? Confused!
I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
- 1,126
65
votes
4
answers
14k
views
Lie derivative vs. covariant derivative in the context of Killing vectors
Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
- 28.6k
59
votes
11
answers
14k
views
Is spacetime wholly a mathematical construct and not a real thing? [closed]
Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
- 649
42
votes
4
answers
8k
views
How do we know that gravity is spacetime and not a field on spacetime?
How do we know that gravity is the curvature of spacetime as opposed to a field, which couples equally to all objects, on spacetime?
- 2,310
39
votes
5
answers
47k
views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
- 929
38
votes
8
answers
5k
views
Interval preserving transformations are linear in special relativity
In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity:
Prove ...
- 3,802
35
votes
4
answers
8k
views
How can we recover the Newtonian gravitational potential from the metric of general relativity?
The Newtonian description of gravity can be formulated in terms of a potential function $\phi$ whose partial derivatives give the acceleration:
$$\frac{d^2\vec{x}}{dt^2}=\vec{g}=-\vec{\nabla}\phi(x)=\...
- 1,436
35
votes
2
answers
5k
views
Why isn't the GPS location calculated from the Schwarzschild metric?
The GPS uses the flat space light propagation formula to calculate the distance from the source (the satellite) to the receiver (observer on Earth):
$$ d=c \cdot \Delta t$$
where $c$ is the speed of ...
- 535
34
votes
11
answers
12k
views
Is the "spacetime" the same thing as the mathematical 4th dimension?
Is the "spacetime" the same thing as the mathematical 4th dimension?
We often say that time is the fourth dimension, but I am wondering if it's means that time is like the fourth geometrical axis, or ...
- 1
33
votes
5
answers
4k
views
Can Lagrangian be thought of as a metric?
My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
- 34.9k
32
votes
3
answers
10k
views
Can quaternion math be used to model spacetime?
Quaternions are commonly used to model 4 dimensional systems where the quaternion consists of a real 3 dimensional vector and an imaginary scalar. So on the surface Quaternions seem well suited to ...
- 11.7k
32
votes
4
answers
4k
views
Can general relativity be completely described as a field in a flat space?
Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
- 11.3k
28
votes
4
answers
4k
views
How do we know the Schwarzschild solution contains an object of mass $M$?
The Schwarzschild metric is
$$ds^2 = - \left( 1 - \frac{2GM}{r} \right) dt^2 + \left(1-\frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\Omega^2.$$
In all GR books, it is stated that $M$ is the mass of the black ...
- 105k
27
votes
4
answers
6k
views
Are black holes perfect spheroids?
What I know about black holes (correct me if I'm wrong) is that they are the most compact objects in the universe that have been discovered. Due to all that gravity, wouldn't black holes be a perfect ...
- 567
26
votes
7
answers
8k
views
Why do we select the metric tensor for raising and lowering indices?
Any rank 2 tensor would do the same trick, right?
Then, which is the motivation for choosing the metric one?
Also, if you help me to prove that $g^{kp}g_{ip}=\delta^k_i$, I would be thankful too ^^...
- 653
26
votes
9
answers
5k
views
Time is the only dimension that has an arrow, and the only dimension which contributes an opposite sign to the metric. Is that just a coincidence?
Time is different from space in these two seemingly independent ways.
One of them is generally believed to have to do with special boundary conditions at the beginning of time.
But if you knew nothing ...
- 1,871
25
votes
5
answers
8k
views
Why do we need tensors in modern physics?
I am wondering why do we really need the concept of tensor. I think it is like vectors, just as a notation of a set of related parameters. I could write the Navier–Stokes equations with scalars, or ...
- 47
25
votes
2
answers
15k
views
Lorentz invariance of the Minkowski metric
As far as I understand, one requires that in order for the scalar product between two vectors to be invariant under Lorentz transformations $x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\,\...
- 3,093
25
votes
2
answers
8k
views
How does the Hubble parameter change with the age of the universe?
How does the Hubble parameter change with the age of the universe?
This question was posted recently, and I had almost finished writing an answer when the question was deleted. Since it's a shame to ...
- 363k
24
votes
5
answers
15k
views
How do I calculate the perturbations to the metric determinant?
I am trying to calculate $\sqrt{-g}$ in terms of a background metric and metric perturbations, to second order in the perturbations. I know how to expand tensors that depend on the metric, but I don't ...
- 541
24
votes
4
answers
14k
views
Why is the covariant derivative of the determinant of the metric zero?
This question, metric determinant and its partial and covariant derivative,
seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
- 749
24
votes
1
answer
2k
views
Causality and how it fits in with relativity
I was talking to my teacher the other day about Einstein's spacetime and there's one thing he couldn't explain about the nature of Cause. I may be being stupid or just unable to comprehend, thanks for ...
- 275
23
votes
5
answers
3k
views
Curvature of Hilbert space
That may appear as a dumb question, but:
Does Hilbert space have curvature, or is it a flat space? How and why?
- 3,735
22
votes
9
answers
11k
views
Minkowski Metric Signature
When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ ...
- 1,049
22
votes
5
answers
8k
views
Inverse and Transpose of Lorentz Transformation
I've seen this question asked a few times on Stack Exchange, but I'm still quite confused why the following "contradiction" seems to arise.
By definition:
$(\Lambda^T)^{\mu}{}_{\nu} = \...
- 736
22
votes
3
answers
9k
views
What is meant when it is said that the universe is homogeneous and isotropic?
It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise ...
- 3,439
22
votes
6
answers
4k
views
Interpretation of a singular metric
I'm interested to find out if we can say anything useful about spacetime at the singularity in the FLRW metric that occurs at $t = 0$.
If I understand correctly, the FLRW spacetime is a combination ...
- 363k
22
votes
2
answers
8k
views
How do I derive the Lorentz contraction from the invariant interval?
While reviewing some basic special relativity, I stumbled upon this problem:
From the definition of the proper time:
$$c^2d\tau^2=c^2dt^2-dx^2$$
I was able to derive the time dilation formula by using ...
- 16.5k
22
votes
5
answers
16k
views
How to prove that orthochronous Lorentz transformations $O^+(1,3)$ form a group?
Orthochronous Lorentz transform are Lorentz transforms that satisfy the conditions (sign convention of Minkowskian metric $+---$)
$$ \Lambda^0{}_0 \geq +1.$$
How to prove they form a subgroup of ...
- 5,042
21
votes
3
answers
4k
views
Why is spacetime not Riemannian?
I apologize if this is a naïve question. I'm a mathematician with, essentially, no upper-level physics knowledge.
From the little I've read, it seems that spacetime is Lorentzian. Unfortunately, the ...
- 467
21
votes
2
answers
10k
views
Covariant vs contravariant vectors
I understand that, in curvilinear coordinates, one can define a covariant basis and a contravariant basis. It seems to me that any vector can be decomposed in either of those basis, thus one can have ...
- 1,016
21
votes
4
answers
2k
views
Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?
My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
- 2,778
20
votes
9
answers
6k
views
Quaternions and 4-vectors
I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity:
$$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$
Is it ...
- 2,930
20
votes
3
answers
22k
views
D'Alembertian for a scalar field
I have read that the D'Alembertian for a scalar field is
$$
\Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu).
$$
Exactly when is this correct? Only for $...
- 15.3k
20
votes
1
answer
8k
views
What do the off-diagonal elements of the metric tensor represent?
For certain metrics in general relativity, the metric tensor $g_{{\alpha}{\beta}}$ is not a diagonal matrix. For example, the Alcubierre metric is given by
$$ds^2 = -dt^2 + [dx - V_s(t) f(r_s) dt]^2 +...
- 493
19
votes
5
answers
2k
views
Where is the Lorentz signature enforced in general relativity?
I'm trying to understand general relativity. Where in the field equations is it enforced that the metric will take on the (+---) form in some basis at each point?
Some thoughts I've had:
It's baked ...
- 193
19
votes
2
answers
2k
views
How can one obtain the metric tensor numerically?
I am self-studying General Relativity.
Is there a method for obtaining the metric tensor exterior to a specified mass distribution numerically? In the simplest case of a spherical mass this should ...
- 1,741
19
votes
3
answers
3k
views
How does one measure space-like geodesics? Or: What is the physical interpretation of space-like geodesics?
In general relativity, time-like geodesics are the trajectories of free-falling test particles, parametrized by proper time. Thus, they are easy to interpret in physical terms and are easy to measure (...
- 475
18
votes
2
answers
18k
views
Lowering/raising metric indexes
So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing.
Consider Minkowski spacetime. The trace of a matrix $A$ can be ...
- 4,113
18
votes
5
answers
21k
views
Why is the space-time interval squared?
The space-time interval equation is this:
$$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$
Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various ...
- 2,182
18
votes
5
answers
2k
views
In general relativity, are two pseudo-Riemannian manifolds physically equivalent if they are isometric, or just diffeomorphic?
In Carroll's Appendix B, he says
You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with ...
- 49.4k
18
votes
2
answers
899
views
Metric of an Evaporating Black Hole
The famous Hawking calculation is done with an assumption that the background is static, i.e. the evaporation doesn't change the mass parameter in the metric. Thus, we simply describe the geometry ...
user87745
17
votes
9
answers
3k
views
"Gauge Freedom" in GR
When we derive the equations for propagating waves in GR, we have to make a gauge choice to get something unique. I understand that in electromagnetism, the gauge is not in general something ...
- 1,657
17
votes
3
answers
7k
views
Deriving Birkhoff's Theorem
I am trying to derive Birkhoff's theorem in GR as an exercise: a spherically symmetric gravitational field is static in the vacuum area. I managed to prove that $g_{00}$ is independent of $t$ in the ...
- 2,926
17
votes
3
answers
4k
views
Advantages of using different metric signatures in relativity and QFT
I am studying General Relativity and some basic QFTs. It bothers me a lot that different books use different metric signatures, i.e. $(-+++)$ and $(+---).$ Can anyone tell me the advantages of using ...
- 171
17
votes
3
answers
3k
views
Proving invariance of $ds^2$ from the invariance of the speed of light
I've started today the book of Landau and Lifshitz Vol.2: The Classical Theory of Fields $\S 2$. They start from the invariance of the speed of light, express it as the fact that $$c^2(\Delta t)^2-(\...
- 171