All Questions
Tagged with spacetime metric-tensor
105
questions
37
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 ...
- 3,732
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 ...
- 16.4k
70
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 ...
- 2,163
105
votes
7
answers
128k
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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 ...
- 36.4k
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
12
votes
4
answers
4k
views
Minkowski spacetime: Is there a signature (+,+,+,+)?
In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if ...
- 3,155
4
votes
2
answers
619
views
Can we just take the underlying set of the spacetime manifold as $\mathbb{R^4}$ for all practical purposes?
In mathematical GR and also in some informal GR presentations (eg: MTW), manifolds are always mentioned before talking about GR... but now I am starting to wonder.. if it even actually neccesary?
In ...
- 7,980
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
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 $$c^2(\Delta t)^2-(\...
- 171
12
votes
2
answers
1k
views
How much choice did Einstein have in choosing his GR equations?
General relativity was summarised by Wheeler as "Spacetime tells matter how to move; Matter tells spacetime how to curve". I have a fairly good mental picture of how the first part works.
However, I ...
- 34.3k
11
votes
7
answers
10k
views
Why there is a minus in the definition of the Minkowski Spacetime Interval?
The spacetime interval is defined as follows:
$$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$
or in tensor notation:
$$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$
...
- 706
5
votes
2
answers
2k
views
Does a vacuum solution to the Einstein equation imply flat spacetime?
I have read that a solution to the vacuum Einstein equation has a vanishing Einstein tensor, and therefore a vanishing stress-energy tensor. This means that there is no matter to generate spacetime ...
- 661
4
votes
3
answers
5k
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What is the physical meaning of the Eddington-Finkelstein coordinates?
What is the physical meaning of the Eddington-Finkelstein coordinates? I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this ...
- 5,794
3
votes
1
answer
483
views
Kleppner derivation of Lorentz transformation
I am reading Kleppner.(Lorentz transformations)
He said,we take the most general transformation relating the coordinates of a given event in the two systems to be of the form
$$x'=Ax +Bt,
y'=y,
z'=z,
...
- 3,455
3
votes
5
answers
3k
views
Time paradox inside a black hole
At the event horizon of a black hole, time and the spatial direction toward the center exchange places. The direction inside the black hole from the event horizon to the the singularity in the center ...
- 12.7k