A Diffeo-invariant action for a matter covector field

ergospherical
Science Advisor
Homework Helper
Education Advisor
Insights Author
Messages
1,097
Reaction score
1,384
I just need a hint to get started, and then I reckon the rest will follow...
We consider a theory where matter is a covector field ##\omega_a## which is described by a diffeomorphism-invariant action ##S_m##. Define:$$E^{a} = \frac{1}{\sqrt{-g}} \frac{\delta S_m}{\delta \omega_a}$$Also, ##T^{ab} = \tfrac{2}{\sqrt{-g}} \tfrac{\delta S_m}{\delta g_{ab}}## is defined as per usual. We would like to derive:$$\nabla_a {T^a}_b = E^a \nabla_b \omega_a - \nabla_a(E^a \omega_b)$$The statement that ##S_m## is diffeomorphism invariant: I guess this means, under ##x\mapsto x - \xi##, and therefore$$\delta g_{\mu \nu} = (L_{\xi} g)_{\mu \nu} = \nabla_{\mu} \xi_{\nu} + \nabla_{\nu} \xi_{\mu}$$that ##S_m## is invariant... how do I use that? I imagine you can write something down in terms of the Lie derivative, and then manipulate the resulting equation into the result. But how to start??
 
Last edited:
Physics news on Phys.org
Never mind, I figured it out. For sure, you write out$$\delta S_m = \int d^4 x \left[ \frac{\delta S_m}{\delta \omega_a} \delta \omega_a + \frac{\delta S_m}{\delta g_{ab}} \delta g_{ab} \right]$$and then stick in the expressions ##\delta \omega = L_{\xi} \omega## and ##\delta g = L_{\xi} g## in terms of covariant derivatives, as well as replacing ##\delta S_m / \delta g_{ab}## by the expression involving the energy momentum tensor. Then just integration by parts gives what you want, when you make sure it vanishes under arbitrary ##\xi##.

Please delete or close the thread if you want... (classic rubber-duck example - write something down and then figure it out)
 
Last edited:
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top