How would you all propose interpreting a claim like "we know that in Schwarzschild spacetime, coordinate time ##t## is the same as proper time ##\tau## as measured by a distant stationary observer."?
This is stated shortly after the other quote I mentioned above. It's the sort of claim that's...
Fantastic, this is what I was looking for as a sanity check - thank you.
As I suggested in my last post, I may very possibly just be misreading the text, but regardless I'm glad I'm not totally off base about my (possibly mistaken) interpretation of it being problematic.
Right, and that would have nothing to do with that observer assigning physical meaning to the difference in coordinate time between the light being emitted and the light being received, right?
I think this is the crux of the question I'm trying to ask. You could use the coordinate time...
I understand that, my apologies if I wrote something that was unclear. This is why I was confused when the text I'm reading seemed to imply that the coordinate time difference I described, between events that are spatially separated, could be understood to correspond to some proper time...
Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?
If so, the question I posed is about a specific case of that: the text I'm reading...
Edit: I'm leaving the original post as is, but after discussion I'm not confused over coordinate time having a physical meaning. I was confused over a particular use of a coordinate time difference to solve a problem, in which a particular coordinate time interval for a particular choice of...
Thanks for the clarification, that does make sense.
Right, that makes sense. So we don't need something as strong as identification of basis vectors at neighboring points in order to define what it is for a tensor to change from point to point.
Thanks again for all your responses, they've...
Thanks again!
So a simple example I was thinking about was the unit circle in polar coordinates. Being a 1D manifold, it only has a single connection coefficient, correct? This is ##\Gamma^{\phi}{}_{\phi\phi} = 0##. This is a statement that the single basis vector ##\vec e_{\phi}## doesn't...
Can I then think of the connection coefficients as defining an identity between the basis vectors in neighboring tangent spaces? My problem earlier with thinking about them in terms of changing basis vectors on a sphere, for example, was that the connection coefficients for polar coordinates on...
Thanks again, all your responses are really helpful. I feel like I've got a better grip on this now.
One last thought, in line with my hope at the onset of this thread of having a working geometrical intuition about the connection coefficients: considering how the connection works on a...
Very helpful, thank you. With your help I think I've figured out where my confusion was originating. I'm pretty sure I was smuggling over some intuition from tangent spaces defined using a position vector, and picturing 2D manifolds as being embedded in 3D Euclidean space (or at least an affine...
This is a really helpful thought, thank you. And my apologies for not understanding earlier!
I think this may resolve a related confusion I've been having with regards to parallel transport as well. If a geodesic is defined as a curve that parallel transports its own tangent vector, I had...
Thanks for this! It makes sense to me how a connection is established in an embedded manifold, such as the surface of a sphere in 3D Euclidean space. I can look at the ##\vec e^{}{}_{\phi}## and ##\vec e^{}{}_{\theta}## basis vectors and note how they change with the coordinates with reference...
So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...
My understanding is that if you live on the surface of the sphere (2D manifold?) then a circle drawn on the sphere will indeed be a circle, but its circumference will be less than ##2\pi r## where ##r## is measured from the center of the circle within the manifold. So the center of the circle as...
I was just reading an intro text about GR, which considers the circumference of a circle on a sphere of radius R as an example of intrinsic curvature - the thought being that you know you're on a 2D curved surface because the circle's circumference will be less than ##2\pi r##. They draw a...
Is this the book pictured in your avatar?
This phrasing is particularly helpful, thank you!
This makes a lot of sense. Thanks again for all of your help, it's really made things much clearer for me!!
This is very helpful, thank you!
I thought of polar coordinates shortly after posting my question. (And felt dumb for not having thought about them immediately!!) A clear case of when the coordinates at a point aren't the components of a position vector.
This is really helpful, thank you...
Thanks for the reply!!
Ah ok, so I was overlooking the fact that the basis I'm working in is defined by tangent vectors to the coordinate curves (I think?). Can I think of the tangent ##\frac {d\vec x}{ds}## geometrically as being a linear combination of the ##\vec E^{}{}_{a}## coordinate...
I hope I'm asking this in the right place!! I'm making my way through the tensors chapter of the Riley et al Math Methods book, and am being tripped up on their discussion of geodesics at the very end of the chapter. In deriving the equation for a geodesic, they basically look at the absolute...
Hi all,
I'm new here, but stumbled on Physics Forums many times during undergrad, usually in desperate search of homework help during fits of "I'm never going to survive this degree." I've been studying on my own to prep for a PhD, but having no one to bounce questions off of is tough and I...