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I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards, Marius

EDIT: In response to the comments which has come in.

Example: Suppose that you want to parallel transport a vector $V^\mu$ on $S^2$, Carroll writes up the equation of parallel transort $$ 0 = \frac{d}{d \lambda} V^\mu + \Gamma^\mu_{\sigma\rho} \frac{dx^\sigma}{d\lambda} V^\rho $$ According the Carroll, this gives me a DE for the continuation for the vector. However, the type of DE that I obtain is dependent on whether $\Gamma^\mu_{\sigma\rho}$ is a constant with respect to $\lambda$, or not. By constant, I mean that there exist an $a^\mu_{\sigma\rho} \in \mathbb{R}$ such that $\Gamma^\mu_{\sigma\rho}(x^\nu(\lambda)) = a^\mu_{\sigma\rho}$ for all $\lambda$. An example of non constant $\Gamma^\mu_{\sigma\rho}$, in contrast, is $\Gamma^\mu_{\sigma\rho}\circ x^\nu$ is an element of the set of smooth functions on the reals with non-compact support. Initially I thought he meant the latter.

Through trail and error I discovered that what he means is that $\Gamma^\mu_{\sigma\rho}$ is actually $\Gamma^\mu_{\sigma\rho}(a)$, where $a$ is the point at which the initial value of $V^\mu$ is specified.

I am taking an introductory course in general relativity, and we are following Sean Carroll's book Spacetime and Geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the equations? Often it does not become clear to me from the context.

For example, suppose that you want to parallel transport a vector $V^\mu$ on $S^2$, Carroll writes up the equation of parallel transport $$ 0 = \frac{\mathrm d}{\mathrm d \lambda} V^\mu + \Gamma^\mu_{\sigma\rho} \frac{\mathrm dx^\sigma}{\mathrm d\lambda} V^\rho $$ According the Carroll, this gives me a DE for the continuation for the vector. However, the type of DE that I obtain is dependent on whether $\Gamma^\mu_{\sigma\rho}$ is a constant with respect to $\lambda$, or not. By constant, I mean that there exist an $a^\mu_{\sigma\rho} \in \mathbb{R}$ such that $\Gamma^\mu_{\sigma\rho}(x^\nu(\lambda)) = a^\mu_{\sigma\rho}$ for all $\lambda$. An example of non constant $\Gamma^\mu_{\sigma\rho}$, in contrast, is $\Gamma^\mu_{\sigma\rho}\circ x^\nu$ is an element of the set of smooth functions on the reals with non-compact support. Initially I thought he meant the latter.

Through trial and error, I discovered that what he means is that $\Gamma^\mu_{\sigma\rho}$ is actually $\Gamma^\mu_{\sigma\rho}(a)$, where $a$ is the point at which the initial value of $V^\mu$ is specified.

I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards, Marius

EDIT: In response to the comments which has come in.

Example: Suppose that you want to parallel transport a vector $V^\mu$ on $S^2$, Carroll writes up the equation of parallel transort $$ 0 = \frac{d}{d \lambda} V^\mu + \Gamma^\mu_{\sigma\rho} \frac{dx^\sigma}{d\lambda} V^\rho $$ According the carroll, this gives me a DE for the continuation for the vector. However, the type of DE that I obtain is dependent on whether $\Gamma^\mu_{\sigma\rho}$ is a constant with respect to lambda, or not. Through trail and error I discovered that what he means is that $\Gamma^\mu_{\sigma\rho}$ is actually $\Gamma^\mu_{\sigma\rho}(a)$, where $a$ is the point at which the initial value of $V^\mu$ is specified.

I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards, Marius

EDIT: In response to the comments which has come in.

Example: Suppose that you want to parallel transport a vector $V^\mu$ on $S^2$, Carroll writes up the equation of parallel transort $$ 0 = \frac{d}{d \lambda} V^\mu + \Gamma^\mu_{\sigma\rho} \frac{dx^\sigma}{d\lambda} V^\rho $$ According the Carroll, this gives me a DE for the continuation for the vector. However, the type of DE that I obtain is dependent on whether $\Gamma^\mu_{\sigma\rho}$ is a constant with respect to $\lambda$, or not. By constant, I mean that there exist an $a^\mu_{\sigma\rho} \in \mathbb{R}$ such that $\Gamma^\mu_{\sigma\rho}(x^\nu(\lambda)) = a^\mu_{\sigma\rho}$ for all $\lambda$. An example of non constant $\Gamma^\mu_{\sigma\rho}$, in contrast, is $\Gamma^\mu_{\sigma\rho}\circ x^\nu$ is an element of the set of smooth functions on the reals with non-compact support. Initially I thought he meant the latter.

Through trail and error I discovered that what he means is that $\Gamma^\mu_{\sigma\rho}$ is actually $\Gamma^\mu_{\sigma\rho}(a)$, where $a$ is the point at which the initial value of $V^\mu$ is specified.

I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards, Marius

I am taking an introductory course in general relativity, and we are following the Carroll book called spacetime and geometry. I've found that he writes his tensor equations without specifying at which point the tensors are to be evaluated. For example, he writes $g_{\mu \nu}$ and not $g_{\mu \nu}(x)$ or $g_{\mu \nu}(x_0)$. I find myself constantly making mistakes because I don't know how to apply the equations without knowing how they are evaluated.

My question: How do I know how to evaluate tensors, in particular the metric tensor or the Christoffel symbol when applying the quations? Often it does not become clear to me from the context.

Thank you for your time.

Kind regards, Marius

EDIT: In response to the comments which has come in.

Example: Suppose that you want to parallel transport a vector $V^\mu$ on $S^2$, Carroll writes up the equation of parallel transort $$ 0 = \frac{d}{d \lambda} V^\mu + \Gamma^\mu_{\sigma\rho} \frac{dx^\sigma}{d\lambda} V^\rho $$ According the carroll, this gives me a DE for the continuation for the vector. However, the type of DE that I obtain is dependent on whether $\Gamma^\mu_{\sigma\rho}$ is a constant with respect to lambda, or not. Through trail and error I discovered that what he means is that $\Gamma^\mu_{\sigma\rho}$ is actually $\Gamma^\mu_{\sigma\rho}(a)$, where $a$ is the point at which the initial value of $V^\mu$ is specified.