Hi, I'm doing the master in science and one of things that I have to study is the Gödel metric. His paper have a high level for me and I'm seeking theses and dissertations about the Gödel universe. At moment I got three theses about the subject in english and portuguese-BR and one dissertation...
Homework Statement
A sequence \{x_{n}\} of real numbers is called bounded if there is a number M such that |x_{n}| ≤M for all n. Let X be the set of all bounded sequences, show that
d(\{x_{n}\},\{y_{n}\})=sup \{|x_{n}-y_{n}| :n \in N \} is a metric on X.The only part I am struggling with is the...
Hi
In the Schwarzschild metric, the proper time is given by
c^{2}dτ^{2} = (1- \frac{2\Phi}{c^2})c^2 dt^2 - r^2 dθ^2
with where \Phi is the gravitational potential. I have left out the d\phi and dr terms.
If there is a particle moving in a circle of radius R at constant angular velocity ω...
How do you find the inverse of metric tensor when there are off-diagonals?
More specifivally, given the (Kerr) metric,
$$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$
we have the metric tensor;
$$ g_{\mu \nu} =...
Hi, could anyone help me out?
The FLRW metric in spherical coordinates is:
\;\; ds2 = dt2 - a(t)2(dr2 + r2dΩ2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)
I am considering a similar metric of the format:
\;\; ds2 = \frac{1}{a(t')^{2}}dt'2 - a(t')2(dr2 + r2dΩ2)...
Hi,
In his original paper, Schwarzschild set the "'equation of the determinant" to be: |g|=-1. In other words, he imposed the determinant of the metric to be equal to minus one when solving the Einstein's equations. Must we impose this equality systematically in general relativity and why...
Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...
Please take a look at the proof I added, there are some things I do not understand with this proof.
1. Does it really show that |f(x)-f(y)|≤d(x,y) for all x and y? Or does it only show that if there is an ball with radius r around x, and this ball is contained in an O in the open covering, and...
I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...
I'm trying to clarify for myself the relation between the stress-energy tensor and the mass scalar term in metric solutions to Einstein's equations. Maybe I should also say I'm trying to understand the energy tensor better, or how it relates to boundary conditions on the solutions.
My...
In another thread WannabeNewton mentioned:
and gave this reference:
Until WBN mentioned it, I had never given any thought to the difference between these methods of measuring rotation, so I would like to explore those ideas further here, particularly in relation to the Kerr metric.
Consider...
Hi so I was just wondering if the metric g=diag(-e^{iat},e^{ibx},e^{icy}) (where a,b,c are free parameters and t,x,y are coordinates) corresponds to a complex manifold (or is nonsensical), and what the manifold looks like?
Homework Statement .
Prove that a closed subset in a metric space ##(X,d)## is the boundary of an open subset if and only if it has empty interior.
The attempt at a solution.
I got stuck in both implications:
##\implies## Suppose ##F## is a closed subspace with ##F=\partial S## for some...
Homework Statement .
Let ##(X_n,d_n)_{n \in \mathbb N}## be a sequence of metric spaces. Consider the product space ##X=\prod_{n \in \mathbb N} X_n## with the distance ##d((x_n)_{n \in \mathbb N},(y_n)_{n \in \mathbb N})=\sum_{n \in \mathbb N} \dfrac{d_n(x_n,y_n)}{n^2[1+d_n(x_n,y_n)]}##...
So, by accident, while deriving the induced metric for a sphere in 3 dimensions I realized that the transpose of the jacobi matrix multiplied by the jacobi matrix (considering it as 3 row/column vectors)will work out the induced metric. Why is it that i≠j ends up being superfluous. One would...
Homework Statement
Show that ##d(f,g) = \int_{0}^{1}\left | f(x) - g(x) \right | dx## is a distance function. Where ##f : [0,1] \rightarrow R## and ##f## is continuous.
Homework Equations
The Attempt at a Solution
I am stuck on the second property where you have to show d(f,g) =...
Homework Statement .
Let ##(X,d)## be a metric space and let ##D \subset X## a dense subset of ##X##. Suppose that given ##\{x_n\}_{n \in \mathbb N} \subset X## there is ##x \in X## such that ##\lim_{n \to \infty}d(x_n,s)=d(x,s)## for every ##s \in D##. Prove that ##\lim_{n \to \infty} x_n=x##...
Hi all
Firstly this is not a homework question. I just found myself wondering what certain numbers meant while at work and realized that i didnt actually know hence the posting.
I was hoping someone could help explain & confirm a couple of things about units.
1) Given the number...
Technically, this is not a homework question, since I solely seek an answer for self-indulgence.
Homework Statement
Example 1.1.4. Suppose f and g are functions in a space X = {f : [0, 1] → R}. Does
d(f, g) =max|f − g| define a metric?
Homework Equations
(1) d(x, y) ≥ 0 for all x, y...
Homework Statement
Hey guys.
So here's the problem:
Consider an ordinary 2D flat spacetime in Cartesian coordinates with the line element
ds^{2}=-dt^{2}+dx^{2}
Now consider a non-inertial coordinate system (t',x'), given by
t'=t, x'=x-vt-\frac{1}{2}at^{2}
(1) What is the metric...
Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V.
In a formula for a projection of the Riemann tensor (see the thread "Projection of the Riemann tensor formula") I encountered the term...
Homework Statement
Let ##S\subset E## where ##E## is a metric space with the property that each point of ##S^c## is a cluster point of ##S.## Let ##E'## be a complete metric space and ##f: S\to E'## a uniformly continuous function. Prove that ##f## can be extended to a continuous function...
Are CTCs in the Kerr metric just an artefact of the coordinates used?
This paper http://arxiv.org/abs/gr-qc/0207014 suggests that is the case. In a private message it has been suggested to me that CTCs in a spacetime are an invariant feature so are not removable by a change of coordinate system...
Homework Statement .
Consider the subspace ##U## of the metric space ##(C[0,1],d_∞)## defined as ##U=\{f \in C[0,1] : f(x)≠0 \forall x \in [0,1] \}##. Prove that ##U## is open and find its connected components.
The attempt at a solution.
First I've proved that ##U## is open. I want to...
This is a spin off from another thread:
First there are a couple of mathpages http://mathpages.com/rr/s8-04/8-04.htm and http://mathpages.com/rr/s8-03/8-03.htm that discuss the refractive index model and highlights the differences.
The first obvious objection is that the 'medium' must have...
Hello,
I need help to calculate the length of pipe in copper (d=22mm) to get the water in "kolam" to be 45 degree celcius and the heat pump temp is 55 degree celcius, heat pump water flow is about 1 meter cubic per hours, any simple formulas ?
thank you for help
I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that:
"It is straightforward to show that the coordinate and dual basis vectors
themselves are related...
"ea = gabeb ..."
I have...
Quick question about the metric space axioms, is the requirement that the distance function be positive-semidefinite an axiom for metric spaces?
It seems that it can be proved from the other axioms (symmetry, identity of indiscernibles and the triangle inequality).
BiP
In the equation
ds2=dx2+dy2+dz2-c2*dt2
the units on the RHS are units of distance squared. But it would seem that units for a spacetime metric should somehow be in units which incorporate both space and time units.
Undoubtedly this is an elementary question, but one has to start somewhere...
Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric.
I want to solve for the Kerr-Newman Metric Tensor.
There are a few questions.
1-What is the value for Q in the equation:
##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}##
where
##G=6.674E-20...
Homework Statement .
Let ##B(a,ε) (ε>0)## in a metric space ##(X,d)##. Decide whether this subset of ##(X,d)## is connected or not.
The attempt at a solution.
Well, I know open intervals in the real line are connected. I suppose that an open ball in a given metric space can be imagined...
I am looking for the Metric Tensor of the Reissner–Nordström Metric.g_{μv}
I have searched the web: Wiki and Bing but I can not find the metric tensor derivations.
Thanks in advance!
1. Homework Statement .
Let ##(X,d)## be a complete metric space. Prove that if ##P \subset X## is perfect, then P is not countably infinite.
3. The Attempt at a Solution .
Well, I couldn't think of a direct proof, I thought that in this case it may be easier to assume is countably infinite...
Homework Statement
Prove that if lim n→∞ (p_n ) = p in a metric space then the set of points {p,p_1,p_2, ...,} are closed.
2. Relevant information
The definition of close in my book is "a set is closed if and only if its complementary is open." So I want to prove this by contradiction. I...
Homework Statement .
Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable.
The attempt at a solution.
I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y:
X is separable...
I have a metric g on spacetime and a spatial metric ##\gamma## such that the components of g can be written in matrix form as
$$ g_ {\alpha, \beta} = \begin{pmatrix} g_{00} & g_{0 j} \\
g_{i 0} & \gamma_{ij} \end{pmatrix} $$
where ##i,j = 1,2,3## and ##\alpha = 0,1,2,3##. Now I want to find a...
Given a basis for spacetime ##\{e_0, \vec{e}_i\}## for which ##\vec{e}_0## is a timelike vector. Of these vectors one can make a new basis for which all vectors are orthogonal to ##\vec{e}_0##. I.e. the vectors $$\hat{\vec{e}}_i = \vec{e}_i - \frac{\vec{e}_i \cdot \vec{e}_0}{\vec{e}_0 \cdot...
Homework Statement .
Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous.
The attempt at a solution.
I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is...
Hey guys,
I know you all wish you never had to do all the weird conversions required for our current system of measurement. I know I'd rather convert 17km to m than 17mi to feet or ever inches. I thought that since I know so many people who would rather just do the easy metric conversions...
I'm writing program in which I generate pseudo random expressions with the hope of finding one that closely approximates the given data set. The functions map from R^m (an m-tuple of reals) to a real. What I need is a way to rate the functions by their accuracy. Are there any known methods for...
Hi, I am having problems in constructing a stress-energy tensor representing a constant magnetic field Bz in the \hat{z} direction. The coordinate system is a cylindric {t,r,z,\varphi}. The metric signature is (+,-,-,-).
I ended with the following mixed stress-energy tensor:
Is this...
Homework Statement
I have been trying to understand the actual applications and mathematics behind Einstein's Field Equations. I have watched a two hour long lecture on how they were derived and have pretty much understood it, however I still don't know how to actually "use" Einstein's Field...
Our current version of the Schwarzschild metric is
d\tau^2=(1-r_s/r)dt^2-(1-r_s/r)^{-1}dr^2-r^2d\Omega^2
where c is set to 1, r is the scalar distance, r_s is the 'event horizon' radius, and d\Omega^2=d\theta^2+sin^2\theta d\phi^2.
In Schwarzschild's original paper from 1916 he does not...
So say you have a 2D Riemannian manifold with a metric defined on it and for simplicity let's say its flat. That means there exists a coordinate system for which the metric tensor is the normal Euclidean metric everywhere. However let's say we are using an arbitrary coordinate system with a non...
what if the evolution of the universe over time and the expansion of space a series of binary events such that the same quantity of energy is divided geometrically from one into two, four, eight, sixteen pieces, such that each Planck time is equivalent to one such split and each Planck length is...
Hello PF:
I noticed a thread on PF in which TOM STOER and others were discussing how to calculate the redshift for an arbitrary metric. I need to talk to Tom if he is still on this list.
The question has arisen in an applied physics field whether the following conformally flat metric...
Ok, this should be an easy one but it's driving me nuts. When we take the Lorentz transformations and apply them to x2-c2t2 we get the exact same expression in another frame. I can do this math easily by letting c=1 and have seen others do it by letting c=1 but I have never seen anyone actually...