We got there in the end. Thanks ever so much everyone for your patience. This has been bothering me for years, so it's a relief to get it cleared up. Even in MTW's Gravitation and Wald's General Relativity there's no reference to symmetric bilinear forms.
So when Schutz talks about the scalar product of two vectors he actually means a symmetric bilinear form (aka the Minkowski metric) acting on those two vectors?
martinbn - yes, I saw your post and the definition. I assumed, when you said "what do you get?" that that was a genuine question, ie that you didn't know the answer and were as puzzled as I am. Whoops, please don't take offence! A definition isn't really an answer though, is it? Am I correct in...
So, as I understand the answers given here, ##\vec{e}_{0}\cdot\vec{e}_{0}=-1## is not a scalar product but a symmetric bilinear form. And the definition of the metric $$g_{\mu\nu}=\vec{e}_{\mu}\cdot\vec{e}_{\nu}$$involves not scalar products but a bunch of (please excuse the technical language)...
In Schutz's A First Course in General Relativity (second edition, page 45, in the context of special relativity) he gives the scalar product of four basis vectors in a frame as follows:
$$\vec{e}_{0}\cdot\vec{e}_{0}=-1,$$...
Homework Statement
I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks
Homework Equations
I am trying to derive the curvature...
Yep, as far as physics is concerned “basic misunderstanding” is my default position. However, in my defence I would say that to the uninitiated this is a genuinely confusing problem. The cars' separation ##s## is given by
$$s=\Delta\phi=d\cos\left(vt\right)=d\sin\theta$$and is calculated...
Thanks to all for your help. Who would have thought two cars on a sphere could be so complicated.
Can I now assume (a) there's a consensus that Void's "constant curvature" answer...
My physics is struggling here, so I'm going to resort to a classic “argument by authority”. My example of geodesic deviation on a sphere is similar to the CalTech Exercise 24.10 here:
http://www.pma.caltech.edu/Courses/ph136/yr2004/0424.1.K.pdf
The solution (Answer 23.10) is given here...
But surely (except at the equator) the distance between the geodesics is not measured along geodesics but along circles of constant \theta. Therefore I can't be using geodesic coordinates?
Thanks. I've been away for a few days, but still can't see this. In my example how are the new (normalized) coordinates related to the old (\theta,\phi) coordinates? Also, the original line element is ds{}^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.How would I find the new line element using the...
Sorry to labour this, but how do you get from the Christoffel symbols for a unit sphere \Gamma_{\theta\phi}^{\phi}=\Gamma_{\phi\theta}^{\phi}=\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}},\Gamma_{\phi\phi}^{\theta}= \sin\theta\cos\theta to Christoffel symbols that vanish?
I know that the...
Thanks, I need to think about this.
I still can't see why the Christoffel symbols vanish along geodesics on the surface of a sphere. Surely the Christoffel symbols will only vanish if we're using a flat metric?
What are geodesic normal local coordinates? Are spherical coordinates always geodesic normal local coordinates? I've tried Googling the term, and got something about "applying the exponential map to the tangent space at p". I'm OK with "tangent space" but have no idea what "exponential map" means.