tag:blogger.com,1999:blog-4005750306801645234.post4107719076208175912..comments2024-02-02T20:17:24.676+01:00Comments on Spacetime and Geometry: Exercise 3.04 Paraboloidal coordinatesUnknownnoreply@blogger.comBlogger12125tag:blogger.com,1999:blog-4005750306801645234.post-4005697125573374832023-10-31T13:49:18.576+01:002023-10-31T13:49:18.576+01:00Thanks Jake! You smarten up formulas surrounding t...Thanks Jake! You smarten up formulas surrounding the Latex code with double # or double $. This is what you wrote.<br />I think the confusion about your (42) and (43) comes down to this: vectors themselves are generally covariant, so when we say that a vector V is *really* the components ##V^\mu## summed over the basis vectors ##e_\mu##, that means that if the vector components transform as $$V^{\mu'} = (\partial x^{\mu'} / \partial x^\mu)V^\mu$$, then in order that the vector itself be invariant the basis vectors have to transform in exactly the opposite way from the vector components; i.e. the fact that $$V^\mu e_\mu = V^{\mu'} e_{\mu'}$$ demands that $$e_{\mu'} = (\partial x^\mu / \partial x^{\mu'})$$ so that when you take the sum, the two transformation matrices – one applied to the components and one applied to the basis vectors – will cancel out and leave the sum (the real thing!) unchanged. This would resolve the tension you noticed in footnote 1.<br /><br />That look about right!Georgehttps://www.blogger.com/profile/04824865122846470839noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-58477762902548224422023-10-30T15:15:23.107+01:002023-10-30T15:15:23.107+01:00Hi George, thanks so much for all of these wonderf...Hi George, thanks so much for all of these wonderful solutions and for the work you've put in. As a GR novice these have been incredibly helpful to me in getting a feel for how to solve these kinds of problems. <br /><br />I think the confusion about your (42) and (43) comes down to this: vectors themselves are generally covariant, so when we say that a vector V is *really* the components V^\mu summed over the basis vectors e_\mu, that means that if the vector components transform as V^\mu' = (\partial x^\mu' / \partial x^\mu)V^\mu, then in order that the vector itself be invariant the basis vectors have to transform in exactly the opposite way from the vector components; i.e. the fact that V^\mu e_\mu = V^\mu' e_\mu' demands that e_\mu' = (\partial x^\mu / \partial x^\mu') so that when you take the sum, the two transformation matrices – one applied to the components and one applied to the basis vectors – will cancel out and leave the sum (the real thing!) unchanged. This would resolve the tension you noticed in footnote 1. <br /><br />Does this seem right? (Apologies for my ugly attempt at inline math; I'm not sure how you do it on this website!) Jakenoreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-71600352524739305232023-01-05T13:43:41.269+01:002023-01-05T13:43:41.269+01:00##\hat{u} = \frac{\partial}{\partial u} = \frac{\p...##\hat{u} = \frac{\partial}{\partial u} = \frac{\partial x}{\partial u} \frac{\partial}{\partial x} = \frac{\partial x}{\partial u} \hat{x}##Georgehttps://www.blogger.com/profile/04824865122846470839noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-82040295052143898202023-01-05T05:21:09.080+01:002023-01-05T05:21:09.080+01:00Your (43) is wrong and hence everything that follo...Your (43) is wrong and hence everything that follows is wrong too. It is clearer to write something like \hat{u} = \frac{\partial}{\partial u} = \frac{\partial x}{\partial u} \frac{\partial}{\partial x} = \frac{\partial x}{\partial u} \hat{x}.Petra Axolotlhttps://www.blogger.com/profile/06597951512037995047noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-75305649243857194262021-12-16T14:42:05.207+01:002021-12-16T14:42:05.207+01:00Quite right! And I marked them up in the pdf, but ...Quite right! And I marked them up in the pdf, but I was too lazy to correct them. Chun first pointed this out over a year ago.Georgehttps://www.blogger.com/profile/04824865122846470839noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-39088874344194763772021-12-15T16:05:08.402+01:002021-12-15T16:05:08.402+01:00I think you have some errors in the calculation of...I think you have some errors in the calculation of the Cristoffel symbols, namely the fact that you have non-vanishing symbols even when all of the three indices are different. This should not be the case (in fact these symbols should vanish) since the metric is diagonal; this follows from the previous exercise. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-78858544871503192432020-11-19T14:11:59.654+01:002020-11-19T14:11:59.654+01:00I have sent my handscript to your email.I hope it ...I have sent my handscript to your email.I hope it won't be classified as spam:(Chun_zzjhttps://www.blogger.com/profile/07800618628646518333noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-33511638013183227192020-11-19T10:27:47.220+01:002020-11-19T10:27:47.220+01:00Not clear! I suppose you are at equations (94) and...Not clear! I suppose you are at equations (94) and onwards. I checked a few they seem OK. Do send me your manuscript: george@general-relativity.net. I will check again later today.Georgehttps://www.blogger.com/profile/04824865122846470839noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-75004985000368745092020-11-18T17:41:11.698+01:002020-11-18T17:41:11.698+01:00One mistake:I find that you fixed the down index o...One mistake:I find that you fixed the down index of g_μυ(like g_11 g_22 g_33) at the beginning when you calculated connection Γ.But after you fixed them, some ∂_σg_μυ which should be vanish do not be vanish.This resulted in a miscalculation of the connection Γ.Am I making myself clear? If not, I can show you my manuscript.Chun_zzjhttps://www.blogger.com/profile/07800618628646518333noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-68797593192145810342020-11-18T16:53:19.918+01:002020-11-18T16:53:19.918+01:00Glad you are still at it Chun! Is there a question...Glad you are still at it Chun! Is there a question here? Or have I made another mistake?Georgehttps://www.blogger.com/profile/04824865122846470839noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-25854815994213581822020-11-18T11:34:43.183+01:002020-11-18T11:34:43.183+01:00Hi,George. When i calculated the divergence of vev...Hi,George. When i calculated the divergence of vevtor V,i found that i use two different formulas (3.32)and(3.34) in carroll's to get different results, which confusing me. Then i found that when u calculated the connection, u first fixed the up index of Γ and next u fixed the two down index together of g,like (94) (104) and (114).However, when we choose different μυρ of g, it turns out that the index of g are not fixed, some of them are going to be vanish.Excuse my poor English！Thx u！Chun_zzjhttps://www.blogger.com/profile/07800618628646518333noreply@blogger.comtag:blogger.com,1999:blog-4005750306801645234.post-3511775093747847352020-05-30T09:10:49.514+02:002020-05-30T09:10:49.514+02:00This comment has been removed by the author.MrKrebbiehttps://www.blogger.com/profile/11710062371981958773noreply@blogger.com