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  1. T

    Mysteries of Geometric Optics In MTW Chapter 22

    Thanks for replying vanhees71, I'll see if I can find something about the WKB approximation or singular perturbation theory - it doesn't look like I can get a free download of Sommerfield Vol IV. Maybe I can find something in my ancient copy of the Feynman lectures. Regards Terry W
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    Mysteries of Geometric Optics In MTW Chapter 22

    Thanks TSny, I'd already decided that I didn't need ##\varepsilon## and that I could get to all the results by "remembering" the order of the terms, but your reply is helpful in reassuring me that I wan't missing something crucial. Cheers Terry
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    Mysteries of Geometric Optics In MTW Chapter 22

    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...
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    MTW Exercise 22.7 -- Calculate the law of local energy conservation for a viscous fluid with heat flow

    Hi Etotheipi, Thanks for getting back to me on this. I'll spend a bit of time comparing your working with mine to see if it generates any further ideas I hadn't thought of doing this, I just took ##\partial_{\mu} (nu^{\mu}) = 0 ## I know that the Lorentz frame was used earlier to help...
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    MTW Exercise 22.7 -- Calculate the law of local energy conservation for a viscous fluid with heat flow

    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...
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    Complete set of answers to Schaum's Tensor Calculus

    Thanks for letting me know. Maybe my post will result in an email and perhaps inspire him to reopen his Schaum! Regards TerryW
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    Complete set of answers to Schaum's Tensor Calculus

    Hi JTMetz, I've just come across your post more or less by accident. I have worked my way through Schaum and (I think) have done all the problems and supplementaries. I reckoned that there were loads of errors in the book, mainly bad typesetting and poor proof reading and I sent an email off to...
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    MTW Ex 21.23 Poynting Flux Vector 'out of the air'

    The answer to my question is yes. I found the key to the solution.:smile:
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    MTW Ex 21.23 Poynting Flux Vector 'out of the air'

    ##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...
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    ADM formulation Initial Value Problem data per spacepoint

    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...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter, Looking back, I was trying to use this formula to get to an equation for ##\psi ( r )## using the ##\Gamma ## terms but it didn't get anywhere. To do this I had to (completely unjustifiably ) set ##\frac {\partial g_{ij}}{\partial t} ## = 0 so it is no surprise that it didn't work...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter and TSny FollowingTSny's suggestion of using 21.137 and 21.136, I've produced the required result, so thanks for that! I'm still going to try to get the result by working out R in terms of derivatives of ##\psi## just for the satisfaction of seeing it all come out. One question...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter and TSny, All I've done here is put ##K_{ik} = 0 ## and ##\Gamma_{pik}## with ##\frac{1}{2}(g_{pi,k} + g_{pk,i} - g_{ki,p})## I think I'll go with this idea for the moment. (Using 21.137 looks more complicated!) - I can see that quite a bit of work is going to be required and I won't...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter, I'm still not quite getting this! I've explored the use of the formula: ##N_{i,j} + N_{j,i} - \frac{\partial g_{ij}}{\partial t} = N^p(g_{pi,j} +g_{pj,i} -g_{ij,p} )## derived from 21.67 Using the three components of the metric ##g_{rr}##, ##g_{\theta \theta}## and ##g_{\phi...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter, Thanks for this. I'll go and have a think. Regards Terry W
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    A poor man's way to Schwarzschild Geometry

    Ok, I've processed the differentiation as follows (in detail): ##\big( \frac{\gamma_i{}^k - \delta _i{}^k Tr \gamma}{N}\big)_{|k} = 0## ##-\frac{N_{|k}}{N^2}(\gamma_i{}^k - \delta _i{}^k Tr \gamma) +-\frac{1}{N}(\gamma_i{}^k - \delta _i{}^k Tr\gamma)_{|k} = 0## The first term disappears...
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    A poor man's way to Schwarzschild Geometry

    Hi Peter, I was on Chapter 8 when you last came to my aid! This comes straight from equation (21.67) Regards TerryW
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    A poor man's way to Schwarzschild Geometry

    The left hand side of 21.127 is ##\frac{\gamma _i{}^k - \delta _i{}^k Tr\gamma}{N}## If ##\gamma _{ij} = \frac {1}{2}(N_{i|j} +N_{j|i} - \frac{∂g_{ij}}{∂t}) = NK_{ij} ## and ##K_{ij} = 0## then surely the LHS = 0 doesn't it? TerryW
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    A poor man's way to Schwarzschild Geometry

    Hi Ibix, Thanks for coming to my assistance. I started off by assuming that the first line of the question is telling me that ##K_{ij}= 0##. In which case, (21.127) reduces to ##8\pi T_i{}^n =0##. If ##T^{nn} = \rho## (which seems to be the implication of (21.132)) and ##T_i{}^n...
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    A poor man's way to Schwarzschild Geometry

    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...
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    Dynamic Equations of the ADM Formalism

    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...
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    Initial Value Equations from ADM Formalism - Trying again

    Here is the final piece of my 'derivation', taking the variation of ##-(2N^{|i}\gamma^{\frac{1}{2}})_i## ##-\delta(-(2N^{|i}\gamma^{\frac{1}{2}})_{,i }) = \delta(-(2N^{|i}\frac{(-g)^{\frac{1}{2}}}{N})_{,i})## ##= \delta(+2N^{|i}(-g)^{\frac{1}{2}}\frac{N_{,i}}{N^2} -...
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    Initial Value Equations from ADM Formalism - Trying again

    Now I am going to look at the variation ##\delta((N^i Tr\pi)_{,i})## ##\delta((N^i Tr\pi)_{,i})## = ##\delta((N^k g_{ij} \pi^{ij})_{,k})## ## = \delta((g_{ij}(N^k \pi^{ij})_{,k} + (N^k \pi^{ij})g_{ij, k})## ##= (N^k \pi^{ij})_{,k}\delta g_{ij}## ##= (N^k{}_{|k} - \Gamma^k{}_{mk} N^m)\pi^{ij}...
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    Initial Value Equations from ADM Formalism - Trying again

    For the next variation, I have to start working on the divergence part (so we can't just ignore it??) and take first of all: ##\delta[(-2\pi^{ij}N_j)_{,i}]## = ##\delta[(-2\pi^{kj}N_j)_{,k}]## = ##\delta[(-2\pi^{kj}g_{ij}N^i)_{,k}]## =##\delta[-g_{ij}(2\pi^{kj}N^i)_{,k} - 2\pi^{kj}N^i g_{ij...
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    Initial Value Equations from ADM Formalism - Trying again

    Hi Mathematical Physicist, I might try this at a later date when I have completed the presentation of my workings. I wonder if I am doing this all wrong but you haven't suggested that this is the case, so I'll carry on with the next bit, which is to find the variation of ##2N_i...
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    Initial Value Equations from ADM Formalism - Trying again

    My original post had quite a few looks but didn't get any replies. I am going to reply to myself and continue to set out my workings in the hope that someone else will join the conversation and help me resolve my problem. In the last part, I managed to produce two of the terms in (21.115). This...
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    Initial Value Equations from ADM Formalism - Trying again

    ##\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...
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    Initial Value Equations out of ADM formalism

    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...
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    ADM field Lagrangian for a source-free electromagnetic field

    Hi anyone who is following my ramblings. I've sorted this now! I should have looked further down the page and spotted that MTW give a definition for ##\mathcal{E}^i## in 21.103. Using this in my recast of 21.99 introduces a whole lot of extra terms but they all cancel out, including my...
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    ADM field Lagrangian for a source-free electromagnetic field

    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) +...
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