Normally I would just dismiss the formula, but I found it in two different sources (both particle physics sources though). One book talked about the vacuum bubble expansion of the integral:
\int \frac{1}{[k^2-m^2][(k-p)^2-m^2]}=\int \frac{1}{[k^2-m^2]^2}
-\int \frac{p^2}{[k^2-m^2]^3}...
This is probably a dumb question, but I have a book that claims that if you have a function of the momentum squared, f(p2), that:
\frac{d}{dp^2}f=\frac{1}{2d}\frac{\partial }{\partial p_\mu}
\frac{\partial }{\partial p^\mu}f
where the d in the denominator is the number of spacetime...
The Christoffel symbols for your first metric in R3 are the same as the ones for your second metric in S2, provided you ignore all Christoffel symbols that have a radius index. Ben Niehoff provided a general formula, but is it just coincidence that the Christoffel symbols are the same for the...
Consider a vector field on a sphere given by \vec{V}= \hat{e}_\phi . This vector field is not parallel transported on a circle at any latitude except the equator. Yet your equation above seems to say that it's parallel transported along any path. That is, \nabla_ie_j = 0 implies u^i \nabla_ie_j...
The derivative straddles two different tangent spaces, since it is a difference of vector fields at two different points. For example in uniform circular motion, the derivative of the velocity points towards the center of the circle. Only a tangential acceleration can be written as a linear...
How can the derivative of a basis vector at a point be the linear combination of tangent vectors at that point?
For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial...
When you say that a (nongeodesic) path rotates with respect to a local orthonormal frame, do you mean that you parallel transport the tangent vector 'u' of the path from point A of the old frame to the orthonormal frame of the new point B via adding: -\Gamma^{i}_{jk}u^j dx^k , and compare this...
If you are traveling in a circle on a circle of latitude, and always pointing your arm North, then doesn't your arm always make the same angle with respect to your path (90 degrees)? Isn't that the pictorial definition of parallel transport?
The parallel transport equation is DV^i=0 along a...
If you walk at constant latitude with your arm always sticking towards the North pole, is that parallel transport of your arm?
The equations don't seem to say it is.
The vector field would be \vec{V}=V(\theta)e_\theta . The component of the vector only depends on theta, but at constant...
I prefer the \Lambda 's to have an inverse sign on them, but that's all right.
Basically that equation says two reference frames have the same metric, or that the metric tensor is invariant under Lorentz transformation.
This implies that two observers take dot products in the same way...
Assume k=0. Then if you measure \rho_0, then don't you have H_0? Or vice versa: if you measure H_0 don't you have \rho_0?
That's what seems to be implied by the equation:
H^2 = \frac{8 \pi G}{3}\rho
, so I'm still not sure how to get a_0
I'm a little confused about the density \rho in the equation:
H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}
Measuring \rho at a single instant in time seems easy. But \rho changes with time. The time dependence of \rho is given as...
You've been very helpful and I don't wish to bother you anymore, especially on something you never use :).
I'm taking a "math methods in physics course", and I have an end-of-semester presentation in a few days, and I was going to choose to present two topics: relativistic quantum mechanics on...
The duffell.org notes were really good. You sure know how to find good notes!
Why SO(3,1)? In the duffell.org notes, the structure group for a vector bundle is GL(4).
I guess you can have any group for your structure group, so long as the group maps the fiber onto itself. So for example, the...
I was reading the Carroll notes you mentioned (great notes by the way), and he mentions adding to each point of the manifold an internal vector space. At each point there were already tangent and cotangent spaces, but now he is adding more vector spaces. He calls the collection of all vector...
So if you have a massless particle, say a graviton, which is a 2nd-ranked tensor, or a neutrino which is a spinor, then would the bending be exactly the same as for light which is a vector? I would add scalar particles but I'm unaware of any that are massless.
So somehow the mass=0 free...
Can the bending of light because of gravity be derived from the Maxwell equations written in curved space time, i.e.,
\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}F^{\mu\nu}})=0
In all the examples the bending of light is treated as a massless particle travelling on a light-like geodesic (if I...
Thanks a lot for your responses. They have been very helpful.
I have a question about the terminology here. Usually the term extrinsic refers to the embedding space of a manifold, and intrinsic refers to the manifold itself. But in this context, you are using extrinsic to mean the spacetime...
There seems to be a bit of asymmetry between fermions and bosons here. Bosons and fermions both transform under the Lorentz group SO(4). When adding general relativity with GL(4), bosons in their vector representations can transform under this group rather easily. But fermions in their spinor...
I have a question about the nomenclature of the "spin connection".
To me "spin" implies quantum mechanics.
However, as I understand it, the spin connection is just a part of the covariant derivative of a vector that is written in tetrad (i.e., local frame) components. So isn't the "spin...
My knowledge about this subjected is very limited, but can't you just use the inverse mapping of \phi, and then the pull back of this inverse transformation would be the "natural" form you seek? And doesn't what Flanders say about 0-forms and pull backs apply to k-forms? You always get a natural...
What is the benefit of expressing Maxwell's equation in the language of differential forms? Differential forms seem to be inferior to the language of tensors. Sure you can do fancy things with the exterior derivative and hodge star, but with tensors you can derive those same identities with...