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Aside from being symmetric, are there any other mathematical constraints on the metric?

Great thanks will have a look at Carrol. I'm specifically interested in metrics of closed spaces where boundary points are identified, like a cylinder or tourus. Any extra complications you think ill run into because of strange boundary conditions?

About 10 years ago I worked on a project where I took a mater distribution and numerically solved for spatial curvature. Can this be done in the opposite direction? Can anybody point me to a resource that would allow me to calculate matter distributions when the metric is specified? What are...

Thanks for the help! Using a wedge between the terms in my original expression got me to the answer!

I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...

So am I correct in the following? There are several KG Green's functions (my first eqn, Advanced, Retarded, Feynman) and the one which we choose is my 2nd equation because it is equivalent to the amplitude of a particle traveling into the future and an antiparticle with negative energy traveling...

Thanks! Watching Tobias Osborne's QFT lectures on youtube and following along with David Tong's lecture notes. Supplementing with Sredniki and Zee.

I like the examples of temperature and mass measured as energy and get but one of the confusing things with natural units is that the fundamental constants are unitless. Also it is intuitively clear that temperature and energy should be related quantities as they are essentially the macro and...

If there were some realistic theories of mechanics that could not be represented using forces then I would argue that forces might not be fundamental. But even if forces weren't fundamental they could still be practical to use in in most cases and definitely worth teaching.

Most strongly interacting conformal field theories have no Lagrangian description. If you know the S-Matrix you have completely specified the theory and there isn't a need for a Lagrangian or action. If you can construct realistic field theories without the PLA you might argue that the PLA is...

The logic of the Feynman Propagator is confusing to me. Written in integral form as it is below $$\Delta _ { F } ( x - y ) = \int \frac { d ^ { 4 } p } { ( 2 \pi ) ^ { 4 } } \frac { i } { p ^ { 2 } - m ^ { 2 } } e ^ { - i p \cdot ( x - y ) },$$ there are poles on the real axis. I have seen...

How is it that when using "natural" units we drop the units themselves. I understand that you can arbitrarily change the magnitude of a parameter by choosing a new unit. For example Oliver R. Smoot is exactly 1 smoot tall. However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is...

Thinking of ##\partial _{\mu}\phi ## as an independent vector field that itself varies seemed to help! Thanks haushofer and samalkhaiat!

I guess my question is "why do variations and derivatives commute?"

When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...

What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the...

So we can interpret the partials and differentials differently, thinking of differentials as little vectors and the partials as components of a transformation operator? If this is the right way to think about things, how should I interpret dy/dx? What is the division of 2 vectors?

Yes, you are right, I should have explicitly included a dependence of x in my function.

I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$ Any insight would be greatly...

In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...

Nobody else seems to have anything in mind. Would appreciate hearing what you have used!

Hi everyone, Does anyone know of a good intuitive resource for learning Yang-Mills theory and Fiber Bundles? Ultimately my goal is to gain a geometric understanding of gauge theory generally. I have been studying differential forms and exterior calculus. Thanks!

I have seen the derivation for Unruh radiation for a massless, non-interacting scalar field (Carroll). Are there interesting differences that arise for more realistic standard model cases. For example, what does QCD look like for an accelerating observer? Any papers that detail this would be...

Can you think of the field as living in a product space of the Minkowski and abstract spaces? Do you know of any reference that explains general yang mills theories geometrically in this way?

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...

I am trying to determine what types of field theories have a Lagrangian that is symmetric under an Infinitesimal acceleration coordinate transformations. Does an infinitesimal generator of acceleration exist? How could I go about constructing this matrix?

Can anybody give a natural interpretation of operators and states in the Heisenberg Picture? When I imagine particles flying through space, it seems that the properties of the particles are changing, rather than the position property itself. Is there any way I should be thinking about these...

Thanks guys these links are great!

I am looking to walk trough hawking and beckenstien's arguments for the proportionality of bh entropy to surface area to better understand black hole entropy. Does anybody know where I can find this calculation? I have taken relativity and qft so I am comfortable with this level of difficulty.

Does anybody know where I can find a walkthrough of the derivation of Black hole entropy the way hawking did it? (I'm not worried about deriving from string theory or lqg) I'm looking to follow along to understand the assumptions in the derivation.