Integration in Sean Carroll's "parallel propagator" derivationby Rasalhague Tags: carroll, derivation, integration, parallel propagator, sean 

#1
Nov410, 01:48 PM

P: 1,400

Reading Chapter 3 of Sean Carroll's General Relativity Lecture Notes, I've followed it up to and including eq. 3.38.
[tex]\frac{d}{d\lambda} P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = A^\mu_{\;\;\; \sigma} P^\sigma_{\;\;\; \rho}(\lambda,\lambda_0).[/tex] Here, Carroll writes, "To solve this equation, first integrate both sides: [tex]P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = \delta^\mu_\rho + \int_{\lambda_0}^\lambda A^\mu_{\;\;\; \sigma} (\eta) \; P^\sigma_{\;\;\; \rho}(\eta,\lambda_0) \; d \eta.[/tex] "The Kronecker delta, it is easy to see, provides the correct normalization for [itex]\lambda = \lambda_0[/itex]." I can see that this makes P the identity matrix in that case, but by what algebraic rule is it inserted. This is a definite integral, so shouldn't any constant of integration be cancelled out? Also, I don't understand the iteration procedure that follows. Doesn't the concept of integration already encode such a procedure, taken to a limit? Should I read this as (coordinatedependent?, coordinate independent?) abstract index notation for a matrix equation inside the integral sign, or is each component function integrated separately? Is there a name for this procedure or the subject area that includes it? Can anyone recommend a book or website that explains the mathematical background. Sorry these questions are a bit vague. I'm not really sure what to ask. I wonder if it's related to what Bachman calls cells and chains. Maybe I should read the rest of that chapter first. 



#2
Nov510, 03:46 AM

P: 1,412

Write it as
[tex]\frac{dP(\lambda)}{d\lambda}=A(\lambda)P(\lambda)[/tex] The solution of this differential equation with the initial data [tex]P(\lambda_0)=I[/tex] is [tex]P(\lambda)=I+\int_{\lambda_0}^\lambda A(\lambda)P(\lambda)[/tex] First, check that indeed the initial value data are satisfied and then that the differential equation is also satisfied. Now, add the indices remembering that [tex]\left(\frac{dP(\lambda)}{d\lambda}\right)^\mu_\rho[/tex] is the same as [tex]\frac{dP^\mu_\rho(\lambda)}{d\lambda}[/tex] and similarly on the RHS  matrix entries of the integral are integrals of matrix entries owing to the linearity of the integral. That is how we define derivatives and integrals of vectorvalued functions. 



#3
Nov510, 05:39 PM

P: 1,400

Thanks again, arkajad! I get it now. Holding lambda_zero constant,
[tex]P(\lambda)  P(\lambda_0) = \int_{\lambda_0}^{\lambda} A(\eta) P(\eta) \; \mathrm{d}\eta[/tex] and [tex]P(\lambda_0) = I[/itex] given what the parallel propagator has be defined to do, so [tex]P(\lambda) = I + \int_{\lambda_0}^{\lambda} A(\eta) P(\eta) \; \mathrm{d}\eta.[/tex] Sean Carroll's eq. (3.39) is just this expressed with indices. And there's no ambiguity over the order of operations, because the derivatives and integrals are defined componentwise. 


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