Integration in Sean Carroll's parallel propagator derivation

[Rasalhague]

Integration in Sean Carroll's "parallel propagator" derivation

Reading Chapter 3 of Sean Carroll's General Relativity Lecture Notes, I've followed it up to and including eq. 3.38.

$$\frac{d}{d\lambda} P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = A^\mu_{\;\;\; \sigma} P^\sigma_{\;\;\; \rho}(\lambda,\lambda_0).$$

Here, Carroll writes, "To solve this equation, first integrate both sides:

$$P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = \delta^\mu_\rho + \int_{\lambda_0}^\lambda A^\mu_{\;\;\; \sigma} (\eta) \; P^\sigma_{\;\;\; \rho}(\eta,\lambda_0) \; d \eta.$$

"The Kronecker delta, it is easy to see, provides the correct normalization for $\lambda = \lambda_0$."

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 canceled 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 (coordinate-dependent?, 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.

 

[arkajad]

Write it as

$$\frac{dP(\lambda)}{d\lambda}=A(\lambda)P(\lambda)$$

The solution of this differential equation with the initial data $$P(\lambda_0)=I$$ is

$$P(\lambda)=I+\int_{\lambda_0}^\lambda A(\lambda)P(\lambda)$$

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

$$\left(\frac{dP(\lambda)}{d\lambda}\right)^\mu_\rho$$ is the same as
$$\frac{dP^\mu_\rho(\lambda)}{d\lambda}$$ 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 vector-valued functions.

 

[Rasalhague]

Thanks again, arkajad! I get it now. Holding lambda_zero constant,

$$P(\lambda) - P(\lambda_0) = \int_{\lambda_0}^{\lambda} A(\eta) P(\eta) \; \mathrm{d}\eta$$

and

[tex]P(\lambda_0) = I[/itex]

given what the parallel propagator has be defined to do, so

$$P(\lambda) = I + \int_{\lambda_0}^{\lambda} A(\eta) P(\eta) \; \mathrm{d}\eta.$$

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.