In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.
A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
First of all i tried to follow the textbook. Here they start of by modelling the atom as an harmonic oscilator:
Then they find the solution as:
They neglect the second term as omega_0 >> gamma which also makes good sense so they end up with:
So far so good. After this they state the...
I'd like to plot the normalized convolution of a Gaussian with a Lorentzian (see the definitions in terms of full width half maximum (fwhm) in the attached image). Here is my attempt, but the print statements with np.trapz() do not return 1 in both cases, but rather ##\approx##0.2. I'd also like...
Part a: Using the above equation. I got
$$\psi(x) = \int_{-\infty}^{\infty} \frac{Ne^{ikx}}{k^2 + \alpha^2}dk $$
So basically I needed to solve above integral to get the wave function. To solve it, I used Jordan's Lemma & Cauchy Residue Theorem.
And obtained $$\psi(x) = \frac {N \pi...
Homework Statement
find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom .
Homework Equations
Acos(w0t)/(d2+t2)
The Attempt at a Solution
I tried first to solve directly . used Euler's identity and got...
My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface?
Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...
Hi, how can I prove that any 2-dim Lorentzian metric can locally be brought to the form
$$g=2 g_{uv}(u,v) \mathrm{d}u \mathrm{d}v=2 g_{uv}(-\mathrm{d}t^2+dr^2)$$
in which the light-cones have slopes one?
Thanks!
Hi,
I know this might be a bit dum but I'm currently stuck with this integral.
In this link: http://www.pci.tu-bs.de/aggericke/PC4e/Kap_III/Linienbreite.htm
I know he's doing the right thing, but I really don't understand the integral of a(omega).
How come it is E(1/(i(ω-ω0) -γ) -...
I was reading into traversable wormholes when I came across this definition from Matt Visser; 'If aMinkowski spacetimecontains a compact region Ω, and if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ...
Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind:
Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be:
We can...
Hi everybody,
Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent...
A spacetime is said to be time-orientable if a continuous designation of which timelike vectors are to be future/past-directed at each of its points and from point to point over the entire manifold. [Ref. Hawking and Israel (1979) page 225]
I want to make sure what conditions must hold in...
Can anyone give a laymans explanation of conformal time in relativity? I tried to read Roger Penrose's book but I found it hard to grasp.Thanks in advance .
Also is a Lorentzian manifold different to a conformal manifold? A laymans explanation would also be much apprecitaed.
From the things I've studied till now, the thought came into my mind that all of the known solutions of Einstein's Field Equations are Lorentzian. Is it correct?
And if it is correct, is there something in EFEs that implies all solutions to it should be Lorentzian?
And the last question, do more...
Lorentz believed in the ether, and developed his version of Relativity Theory based on interactions of moving charged particles (electrons, specifically) with the ether. What I don't understand is how one always gets length contraction for movement in an inertial system moving with respect to...
Hi everyone, I am having some problems understanding Bergmann's problems.
Problem 3 from Chapter 4 from Intro to the Theory of Relativity by Bergmann
1. Suppose that the frequency at a light ray is f with respect to a frame of reference S. Its frequency f′ in another frame of reference, S'...
In the recent analysis by Mielczarek, Linsefors, and Barrau, the key parameter is ρc the highest density achieved in the bounce. The extremes (negative and then positive) of the Hubble rate occur symmetrically at density ρc/2
In the contraction preceding bounce, as the density approaches ρc/2...
In another thread, it was asked if we could use the angular deficit idea to determine curvature not in space, but in space-time.
My idea to attempt to proceed along these lines would be to generalize the idea of angle, but I don't have anything that I feel I can point to.
As a starting...
I wasn't exactly paying much attention during our lab, partially because it was right in the middle of midterms and I wanted to solely focus on those. Now it is coming back to bite me as I am not entirely sure how to complete my lab report, or the theory behind the lab really.
If you would like...
Hi
I have a problem where the flux of a particle beam is measured using a (nearly) perpendicular laser beam and a photomultiplier.
I have a function looking like this:
L(\nu) = \frac{\gamma/2}{(\nu - \nu_0 + kv)^2 +(\gamma^2/4)}
I suppose this is a Lorentzian lineshape function...
hi
I know the Fourier transform of a lorentzian function is a lorentzian but i was wondering if the Fourier transform of the second derivation of a lorentzian function is also a second derivative of a lorentzian function
Thanks
Hi
I hope some one can help me with this one:
I have a Lorentzian line profile
L(√L) = 1 / ((√L - √0 )^2 + (\Gamma2/4))
for v = 0.
For v \neq 0 I have
\int(1 / ((√L - √0 - kvz)^2 + (\Gamma2/4)) * g(vz) dvz)
I suppose the factor g(vz) dvz is a velocity factor, but how do I...
Hi
I have the following code:
lorentz[A_, Ox_, Oy_, FWHM_, x_] :=
A (1/3.14) FWHM/((x - Ox)^2 + FWHM^2) + Oy;
gauss[A_, Ox_, Oy_, x_, C_] := A Exp[-(x - Ox)^2 C] + Oy;
Convolve[lorentz[1, 0, 0, 1, x], gauss[1, 0, 0, x, 1], x, y]
It takes extremely long time for this to finish -- is it...
Lorentzian contraction of a box increases the pressure inside?
I have been doing some research on gravity, and I am having a very hard time with the following example I found in a book entitled Gravity From the Ground Up by Bernard Schutz. It sounds like a 3rd grader explaining...
Homework Statement
Resonances occur in many physical systems, and are often observed by measuring the frequency response of the system to an applied driving force. use the example of a damped harmonic oscillator to show how the lorentzian function serves as a good description of resonant...
Hi,
I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian...
Hendrik Lorentz, a Nobel Prize-winning physicist who was a mentor to Einstein, developed his own theory of relativity before Einstein. Einstein's theory uses the "Lorentz transformations" explicitly, but the interpretation of these formalisms is quite different in each theory.
Einstein made...
http://arxiv.org/abs/1012.4216
4-dimensional Spin-foam Model with Quantum Lorentz Group
Muxin Han
22 pages, 3 figures
(Submitted on 19 Dec 2010)
"We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz...
Homework Statement
1) I try to understand the lorentzian lineshape and relate to the gaussian graph but i don't know what is the difference.
2) Uncertainty principle- if it's related to the lorentzian lineshape it will give the information about the lifetime in the well and the width of...
It is fairly easy to prove that each manifold can be given a Riemannian structure. The argument is standard: locally you give the riemannian structure and then you use partions of unity. This proof breaks down for signed metrics. Even for a manifold requiring only two charts. For example, I've...
H.E. Lorentz, a Nobel Prize-winning physicist, was an older contemporary of Einstein's who inspired and guided Einstein in his work. Lorentz, however, never fully subscribed to special or general relativity and believed to his death that what is now called "Lorentz Ether Theory" was the better...
Hi,
I'm trying to understand Lorentzian relativity (Lorentz ether theory, whatever) which is empirically equivalent to the Einsteinian STR. I have, however, a problem in comprehending length contraction.
In the Lorentz theory we have a preferred frame and length contraction is a real...
Matt Visser - Lorentzian wormholes. From Einstein to Hawking (Springer) (1996)
Just opened this book.
Does anybody know it, and if you do, do you recommend it?
Any feedback on this book.