Assuming dark energy is fairly, uniformly distributed through out the cosmos, how strong is it, or how much energy is associate with it, out in the deepest, emptiest voids in space? I'm specificlaly refering to the great voids in between the great walls of galaxy clusters. I'm making the...
This might sound as a dumb and silly question but if you think about it, it makes sense. If we wrongly assume that gravity is a force just like any other, and given the fact that time is closely related to gravity and that gravitational time dilation is a thing, wouldn't reverse time travel...
I was discussing this paper with a couple of physicists colleagues of mine (https://arxiv.org/abs/2011.12970)
In the paper, the authors describe "spacetimes without symmetries". When I mentioned that, one of my friends said that no spacetime predicted or included in the theory of relativity...
I'm not aware of the mathcode here, so forgive me for not posting my work straight away. I simply need to ascertain what code first displays equations.
$a$
a
a
Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
If so, What...
Here is the video: [link deleted by moderators]
His basic idea is to take the spacetime interval and add a 5th term for the 5th dimension he is describing so it looks like: $$\Delta S^2 = c^2\Delta t^2 + c^2\Delta w^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 $$
where w is the difference in time...
Hi, I've just calculated the orbital precession for the earth using the sigma formula of general relativity.
$$
\sigma=\frac{24 \pi^{3} R^{2}}{T^{2} c^{2}\left(1-e^{2}\right)}=\frac{24\pi^3×1.5×10^{11}}{3×10^7×3×10^8(1-0.0034^2)}=0.012
$$
What is the unit of the result? Degrees per century or...
What does and observer inside of a collapsing shell observe? Lets say we have a shell of matter collapsing to a black hole. What would observers near the center see? How would the rest of the universe appear when,
The shell is approaching the Schwarzschild radius?
After the shell passes the...
Any help to understand how the authors of this paper
Fine Tuning Problem of the Cosmological Constant in a Generalized Randall-Sundrum Model
calculted this size of the extra dimension Equ. (3.8) from the scale factor defined by Equ. (3.3) ? Specifically, this paragraph after Equ. (3.8)
-...
From this post-gradient energy in classical field theory, one identifies the term ##E\equiv\frac{1}{2}\left(\partial_x\phi\right)^2## as the gradient energy which can be interpreted as elastic potential energy.
Can one then say that $$F\equiv -\frac{\partial...
In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away)
Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...
In natural units, it’s known that the unit of the cosmological constant is ##eV^2##.
I don‘t get why in this paper :
https://arxiv.org/pdf/2201.09016.pdf
page (1), it says the value of ##\Lambda \sim meV^4##, this means ##\Lambda \sim (10^6 ~ eV)^4 \sim 10^{24} eV^4 ##, shoud not the unit ##eV...
I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##.
The normal vector is given by,
##n^\mu = g^{\mu\nu} \partial_\nu S ##
How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##?
Also, after...
In the second paragraph on page 25 of Wald's General Relativity he rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sese...
Hi!
I want to start with saying that I'm not an expert on these type of problems, but I will be gratefull for some calarifications.
I've heard that there's nothing in psysics that says that time travel is impossible. I want to make a case with the time traveling battery. Could be any mass with...
Some physicists prefer to explain the problem of conservation of energy in General Relativity by considering the gravitational potential energy of the universe that would cancel all the other energies and therefore the energy in the universe would be conserved this way.
However, many other...
When we compute the stress energy momentum tensor ## T_{\mu\nu} ##, it has units of energy density. If, therefore, we know the total energy ##E## of the system described by ## T_{\mu\nu} ##, can we compute the volume of the system from ## V = E/T_{00}##?
If it holds, I would assume this would...
Hello!
I'm starting to study curved QFT and am slightly confused about the invariance of the Klein Gordon Lagrangian under a linear diffeomorphism.
This is $$L=\sqrt{-g}\left(g^{\mu\nu}\partial_\mu \phi \partial_\nu \phi-\frac{m^2}{2}\phi^2\right),$$
I don't see how ##g^{\mu\nu}\to...
Homework Statement:: See below.
Relevant Equations:: See below.
I am trying to calculate the event horizon and ergosphere of the Kerr metric. However, I could not seem to find a proper derivation or formula to calculate the event horizon and ergosphere. Could someone point me to the...
Hello everyone,
I know that GR equations are complicated and beyond my scope.
But does GR give a simple gravitational equation: Force (as we know it) as a function of distance? (without any complicated tensors).
- If yes. What is the equation? Does it give us something similar to Newtons...
I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar.
Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...
Question ##1##.
Consider the following identity
\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\epsilon_{i}^{\phantom{i}lm}=h^{jl}h^{m}_{\phantom{m}k}-h^{jm}h^{l}_{\phantom{l}k}
\end{equation}
which we know holds in flat space. Does this identity still hold in curved space? and if so, how...
In the Reissner–Nordström metric, the charge ##Q## of the central body enters only as its square ##Q^2##. The same is true for the Kerr-Schild form. This would seem to imply that all effects are even functions of ##Q##. For example, the gravitational time dilation is often written as
$$\gamma =...
Consider the following action
$$S=\int\mathrm{d}^3x\sqrt{h}\left[R^{(3)}-\frac{1}{4}\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\chi^{-1}\chi^{,i}\right)\right]$$
where ##h## is the determinant of the 3-dimensional metric tensor ##h_{ij}## and ##R## is the Ricci scalar.
I want to get the equations of...
An exact gravitational plane wave solution to Einstein's field equation has the line metric
$$\mathrm{d}s^2=-2\mathrm{d}u\mathrm{d}v+a^2(u)\mathrm{d}^2x+b^2(u)\mathrm{d}^2y.$$
I have calculated the non-vanishing Christoffel symbols and Ricci curvature components and used the vacuum Einstein...
I want to learn special relativity.I have read a tiny bit of 2nd edition of Spacetime Physics: Introduction to Special Relativity and am liking it. Is it a good book? I also want problems to solve. I tried Special Relativity: For the Enthusiastic Beginner but found it to difficult. Does anyone...
I'm trying to understand how the RS model solved the hierarchy problem from this mass relation
$$ M^2_p = \frac{M^3}{k} \Large[1- e^{-2k\pi r} \Large],$$
Equ. 16 in their paper:
https://arxiv.org/abs/hep-ph/9905221
With k as large as the Planck scale, the exponential will be so small and...
Hello!
The paper I study is related to string theory and modified gravity theories topics.
As they say in page 5 “The four-dimensional effective theory now follows by substituting Eq. (13) into the original action, Eq. (4)”
I wonder how did they drive a 4- dimensional effective metric...
I have been reading the book Spacetime and Geometry by Sean Carroll, especially Ch. 2 Manifolds and Ch. 3 Curvature. I'm just wondering are there any lecture notes or books with lots of practice problems (with solutions or at least answers the better) that is suitable for physicist?
To give an...
It is said that: It is not possible to write a position vector in a curved space time. What is the reason?
How can one describe a general vector in a curved space time?
Can you please suggest a good textbook or an article which explains this aspect?
OK. Gravity is not a force it is a contraction or curvature of space.
I was free-falling and now I hit the ground. Why don't I float through the universe, or go upward instead of still trying to go downward.
Because I hit the ground, and now there is no force(like gravity) and my free-falling...
This question wasn't particularly hard, so I assume metric compatibility and input Ricci tensor to the left side of Einstein's equation.
$$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=Cg_{\mu\nu}-\frac{1}{2} (4C)g_{\mu\nu}=-Cg_{\mu\nu}$$
Then apply covariant derivative on both side...
It can be shown mathematically that the scalar massless wave equation is conformally invariant. However, doing so is rather tedious and muted in terms of physical understanding. As such, is there a physically intuitive explanation as to why the scalar massless wave equation is conformally invariant?
a) Can we convert energy to mass (matter) in every day life?
b) When we charge a phone battery, its mass (weight) increases according to E=mc2 . Does it mean we convert energy to matter? If not, how its mass increases?
I'm reading "Differentiable manifolds: A Theoretical Physics Approach" by Castillo and on page 170 of the book a calculation of the Ricci tensor coefficients for a metric is illustrated. In the book the starting point for this method is the equation given by:
$$d\theta^i = \Gamma^i_{[jk]}...
We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces ##\Sigma## with
\begin{align}
n^\alpha n_\alpha=\epsilon=\pm1
\end{align}
where ##\epsilon=1## for timelike and ##-1## for spacelike. We can define a tensor ## h_{\alpha\beta}## on ##\Sigma## by...
What I've done is using the TOV equations and I what I found at the end is:
##e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)##
so I am sure that this is not right, if someone can help me knowing it I really apricate it :)
Hi everybody
I saw quite a nice Youtube vid about general relativity and how gravity bends spacetime and therefor redirects angular momentum into the center of gravity. I thought the first time I begun to understand the concept but immediatly the questions poped up.
The video basically says...
So say I have a bubble embedded in a spacetime with metric:
$$ds^2 = -dt^2 + a(t) ( dr^2 + r^2 d\Omega^2_2) $$
how do I compute the normal vector if I assume the wall of the bubble the metric represents follows a time-like trajectory, for any ##a(t)##?
Since we are interested in dynamical...
Hi all,
I need help understanding the light ray bending in the original GR 1916 paper, Die Grundlagen....
First of all, Einstein states the ##c## is not an invariant in GR.
In fact, from (70) and (73), it stems that $$\gamma = \sqrt{ -\frac {g_{44}}{g_{22}} }, $$ where ##\gamma## is ##|c| <= 1##...
Via web search found https://www.physicsforums.com/threads/what-dimension-does-space-time-curve-in.852103/
Read it and watched two videos mentioned:
I understand we cannot perceive 5D ;-), so extrinsic visualization of maximum of 2D intrinsic curvature is possible. So time+1d space is all we...
I'm watching the Stanford University Lecture series: Einsten's General Theory of Relativity presented by Leonard Susskind (who incidentally has to be one of the greatest educators I've ever watched).
Whilst deriving the basic divergence equations relating acceleration, mass density, and...
The ansatz for the 5D metric is
\begin{equation}
G_{\mu \nu}= g_{\mu \nu}+ \phi A_{\mu} A_{\nu},
\end{equation}
\begin{equation}
G_{5\nu} = \phi A_{\nu},
\end{equation}
\begin{equation}
G_{55} = \phi.
\end{equation}
This information was extremely enlightening for me, but what's the analogous...
Geonaut
Thread
generalrelaivity
higher dimensions
kaluza-klein theory
1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities
$$Q^t :=...
Hi,
in general relativity I'm aware of the spacetime 'distance' between two timelike related events is maximized by the free falling timelike path (zero proper acceleration) joining them.
Consider now a couple of events belonging to a spacelike hypersurface (AFAIK it is an hypersurface with...
Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie, published in 1916's Annalen Der Physik, I came across some equations which I couldn't verify after doing the computations hinted at.
The first are equations 47b) regarding the gravity contribution to the...