At the start of this section §22.5 (Geometric Optics in curved Spacetime), the amplitude of the vector potential is given as:
A = ##\mathfrak R\{Amplitude \ X \ e^{i\theta}\} ##
The Amplitude is then re=expressed a "two-length-scale" expansion (fine!) but it then is modified further to...
I've come to a grinding halt with this and I can't see a way forward.
Can someone please take a look at what I've done so far and let me know if what I have done is OK and then if it is, give me a hint on how to proceed.
First up,
Is ## u \cdot \nabla \cdot T = u_\alpha...
##4\pi\mathcal L = -\mathcal e \frac{\partial A_i }{\partial t} - \phi\mathcal E^i{}_{,i} -\frac{1}{2}N\gamma^{\frac{1}{2}}g_{ij}(\mathcal E^i \mathcal E^j +\mathcal B^i\mathcal B^j) +N^i [ijk]\mathcal E^i\mathcal B^j## MTW (21.100)
I'm trying to produce the result required by the problem...
I'm having a bit of trouble getting a clear picture of what is going on here, so if anyone can shed any light, it will be greatly appreciated.
1. I can see how the metric coefficients provide the six numbers per spacepoint, but it can't always be possible to transform the metric into a diagonal...
Can anyone help me get started with this problem?
What should I use for Gni?
I've tried to produce Tni by working out Rni (using methods developed in an earlier chapter) but the results don't lead me anywhere.
I'm really stuck for a way forward on this problem so if anyone can help, it...
The main error in my earlier work was forgetting that to obtain ##\delta X## you have to find not only ##\frac {\partial X}{\partial g_{ij}}##, but also ##\big(-\frac {\partial X}{\partial g_{ij,k}}\big)_{,k}## and ##\big(\frac {\partial X}{\partial g_{ij,kl}}\big)_{,kl}##. I also missed a...
##\delta(- N\mathcal{H}) = \delta(-N[\gamma^{-\frac{1}{2}}(Trπ ^2 - \frac{1}{2}(Trπ )^2 -\gamma^{\frac{1}{2}}R])##
and I am going to concentrate on the first part and find ##\delta(-N[\gamma^{-\frac{1}{2}}(Trπ ^2 - \frac{1}{2}(Trπ )^2]) ## wrt ##g_{ij}##.
##\delta(-N[\gamma^{-\frac{1}{2}}(Trπ...
In my earlier post, I demonstrated a way to derive MTW's Equation (21.90),
##16π\mathfrak{L}_{geom} = - \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i\mathcal{H^i}-2(π^{ij}N_j-½N^iTrπ+γ^½N^{|i})_{,i}## MTW (21.90)
I received my first two 'likes' for this which made me really happy - Thanks...
Homework Statement
I am trying to reproduce MTW's ADM version of the field Lagrangian for a source free electromagnetic field:
##4π\mathcal {L} = -\mathcal {E}^i∂A_i/∂t - ∅\mathcal {E}^i{}_{,i} - \frac{1}{2}Nγ^{-\frac{1}{2}}g_{ij}(\mathcal {E}^i\mathcal {E}^i + \mathcal {B}^i\mathcal {B}^i) +...
Homework Statement
This post contains the answer to my thread of 10th August...
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in which I asked if anyone could point out how to derive
##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0 \,_{pq} - g_{pq} ^{(4)} \Gamma ^0\, _{rs} g^{rs}) g^{pq} g^{jq}##
from
##\mathfrak {L}## = (4)R...
Homework Statement
This isn't a request for assistance, I am just posting this to help anyone else in the future who wants to see how MTW's equation 21.90 can be developed from the simple Lagrangian.
MTW's Equation 21.83 is simply ##16π\mathfrak{L}_{geom} = (-^{(4)}g)^{(4)}R##
One page...
Homework Statement
I've been working through a paper by Alexey Golovnev, title 'ADM and massive gravity' arXiv.1302.0687v4 [gr-qc] 26 March 2013. I am hoping to use his result for the Einstein-Hilbert density to achieve my aim of finding a way to derive Equation 21.90 in MTW. I have worked my...
Homework Statement
If you look in Wikipedia for ADM formalism, you are given a Derivation, which starts from the Lagrangian:
##\mathfrak {L}## = (4)R ##\sqrt{^{(4)}g}## and moves rapidly to....
The conjugate momenta can then be computed as
##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0...
Homework Statement
On the attached two pages from MTW, there are two expressions for the variational principle. I've worked my way through to get to (21.86) (+21.88) and have continued to the bottom of the page to (21.90). I then thought I'd try to use (21.91), (21.92) and (21.93) to see if I...
Homework Statement
Occasional, MTW's use of language is a bit obscure. I've highlighted a section from p491 (attached) which I just can't make any sense of.
I know about the 20 components of Rαβγδ but what does "these twenty components are arbitrary to the extent of the six parameters of a...
Homework Statement
I have a couple of issues with this problem, but first of all, I want to find out what I am missing in my attempt to prove ##∇^2Φ = 4πρ##
Homework Equations
See below
The Attempt at a Solution
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If ##Φ = \frac {-M}{r}##
Then ##Φ_{,i} = \frac {Mx^i}{r^3}##
So ##Φ_{,ii}...
Homework Statement
This is Exercise 19.1 in MTW - See attachment
Homework Equations
See attachment
The Attempt at a Solution
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I've worked through 19.3a, 19.3b and 19.3c ( see post by zn5252 back in March 2013 replied to by PeterDonis) and proved them for my my own satisfaction and I've...
Homework Statement
This is Exercise 15.2 in MTW - See attachment
Homework Equations
See attachment
The Attempt at a Solution
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My attempt at a solution is also in the attachment.
Are my initial assumptions OK? If not can someone nudge me in the right direction.
If my initial...
Homework Statement
I've been working on Exercise 14.3 in MTW. This starts with the FLRW metric (see attachment) and asks that you find the connection coefficients and then produce the non-zero elements of the Riemann Tensor.
The answer given is that there are only 2 non-zero elements vis...
I've run into a couple of problems with exercise 12.6 in MTW (attached).
At this stage, all I'm asking is for someone to give me answers to the following questions:
1. What are the Rα'β'γ'δ' in the primed co-ordinate system? and
2. What are the non-zero connection coefficients in the...
I have managed to work out parts a and b of Exercise 13.7 from MTW (attached), but can't see how part c works.
I can see how it could work in (say) the example of taking a radar measurement of the distance to Venus, where we have the Euclidian distance prediction and the result of the radar...
I'm working through Meisner Thorne and Wheeler (MTW), but have been temporarily sidetracked by a problem with rotation matrices.
I've worked through the maths and produced the matrices by multiplying the three individual rotation matrices, (no problem there) but I have been trying to work out...
I have managed to get to the end of Chapter 6 and have done almost all of the exercises (I didn't get anywhere with exercise 5.6 (d) and have seen the Cesarth/TSny exchange and still don't feel I have a satisfactory solution...) but I have hit a bit of a wall with Exercise 7.1 (a).
First...
I have managed to get to the end of Chapter 6 and have done almost all of the exercises (I didn't get anywhere with exercise 5.6 (d) and have seen the Cesarth/TSny exchange and still don't feel I have a satisfactory solution...) but I have hit a bit of a wall with Exercise 7.1. I've attached my...
I've been working on Ex 5.4 in MTW. The maths is fairly straight forward, but I don't really understand what is going on!
In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these...
Can anyone help me with Ex 4.1 in MTW?
What is a 'generic' field? My expectation is that it would comprise an Electric Field, with arbitrary direction, a Magnetic Field, also with arbitrary direction, and a radiation field (E and B of equal magnitude) , radiating in an arbitrary direction...
I have spent much effort trying to prove that det|\Lambda\mu\upsilon| = +1 or -1 (following a successful effort to prove (3.50g) on p87 of MTW)
From the result of \LambdaT\eta\Lambda = \eta I've produced four equations like:
\Lambda00\Lambda00 - \Lambda01\Lambda01 - \Lambda02\Lambda02 -...
If the squared length of a 4-velocity is -1, how can you have a component v0 =0?
I've played with equation (2.35) and produced a result of v = γ(-u2, -u1, -u2, -u3), which doesn't have a squared length of -1.
Can anyone help out?
TerryW