Derivative of Christoffel symbols in a local inertial frame

Question

I have a doubt regarding the Riemann tensor in a LIF. The general expression of the Riemann tensor is:

$R^{\alpha}_{\beta \mu \nu} = \Gamma ^{\alpha}_{\beta \nu, \mu} - \Gamma ^{\alpha}_{\beta \mu, \nu} -\Gamma ^{\alpha}_{\kappa \nu} \Gamma ^{\kappa}_{\beta \mu} +\Gamma ^{\alpha}_{\kappa \mu} \Gamma ^{\kappa}_{\beta \nu} \tag{1}$

where $A_{ \alpha \beta, \mu \nu } = \dfrac{\partial A_{\alpha \beta}}{\partial x^{\mu}\partial x^{\nu}}$

we recognize the first two parts which are linear in the second derivative and the other two parts nonlinear in the first derivatives of metric tensor.

The R. tensor has a very nice form when computed in a Locally Inertial Frame:

$R^{\sigma}_{ \beta \mu \nu} = \dfrac{1}{2}g^{\sigma \alpha}[g_{\alpha \nu, \beta \mu} - g_{\alpha \mu, \beta \nu} +g_{\beta \mu, \alpha \nu} -g_{\beta\mu, \alpha \nu} ] \tag{2} $

We know that in flat spacetime and consequently in a LIF Christoffel symbols vanish. The nonlinear part of $(1)$ is zero, thus we only have the second derivatives of metric tensor i.e. $(2)$ which are related to the derivatives of Christoffel symbols in $(1)$.

The WELL known definition of Local Inertial Frame (or LIF) is a local flat space which is the mathematical counterpart of the general equivalence principle. If we know $g_{\mu\nu}$ and their first derivatives (i.e. $\Gamma^{\alpha}_{\mu \nu}$) in the point $X$, in a general spacetime we can always determine a locally (inertial) frame $\xi^{\alpha}(x)$ in the neighborhood of $X$. From (e.g. {1}) the following expression:

$ \dfrac{\partial ^2 \xi ^{\beta}}{\partial x^{\mu} \partial x^{\nu} } = \dfrac{\partial \xi^{\beta}}{\partial x^{\lambda}} \Gamma^{\lambda}_{\mu \nu} \tag{3}$

we are able to write the series expansion near $X$ up to the second order:

$ \underset{x \approx X }{ \xi^{\beta}(x)} = \xi^{\beta}(X) + [\dfrac{\partial \xi^{\beta}(x)}{\partial x^{\lambda}}]_{x=X} (x^{\lambda}- X^{\lambda}) + \dfrac{1}{2}[\dfrac{\partial \xi^{\beta}(x)}{\partial x^{\lambda}}\Gamma^{\lambda}_{\mu\nu}]_{x=X} (x^{\mu}- X^{\mu}) (x^{\nu}- X^{\nu}) + \text{higher orders} := \\ := a^{\beta} + b^{\beta}_{\lambda}(x^{\lambda}- X^{\lambda}) + \dfrac{1}{2} b^{\beta}_{\lambda} \Gamma^{\lambda}_{\mu\nu}(x^{\mu}- X^{\mu}) (x^{\nu}- X^{\nu}) + \text{higher orders} $

in addition, since it must be a locally flat space we have to relate the old frame to the new via $\eta_{\mu \nu}$ :

$g_{\mu \nu}(X)= \eta_{\alpha \beta} \dfrac{\partial \xi(x) ^{\alpha}}{\partial x^{\mu}}|_{x=X}\dfrac{\partial \xi (x)^{\beta}}{\partial x^{\nu}}|_{x=X} \equiv \eta_{\alpha \beta} b^{\alpha}_{\mu} b^{\beta}_{\nu}$

From the previous equation we find $b^{\beta}_{\mu}$. As regards $a^{\beta}$ there is an ambiguity but we have still the freedom to make a Lorentz transformation and the new frame is still locally inertial.

My question is: in a LIF why are Christoffel symbols equal to zero but their derivatives not?

My possible answer:

If we differentiate the rhs of $(3)$ with respect $x^{\sigma}$ we can use again $(3)$ with with other indices:

$ \dfrac{\partial ^3 \xi ^{\beta}}{\partial x^{\sigma} \partial x^{\mu} \partial x^{\nu} } = \dfrac{\partial \xi^{\beta}}{\partial x^l} \Gamma ^l _{\sigma \lambda} \Gamma^{\lambda}_{\mu \nu} + \dfrac{\partial \xi^{\beta}}{\partial x^s} \Gamma ^{s}_{\mu \nu , \sigma} \tag{4} $

After moving in a LIF (i.e. connections vanishes), from $(4)$ we can isolate $ \Gamma ^{s}_{\mu \nu , \sigma}$ .

What do you think?

{1} Carroll, S. M. (2019). Spacetime and geometry. Cambridge University Press.

Answer 1

I think I have answered this before but one can construct co-ordinates in which $$ g_{\mu\nu}(x)= \delta_{\mu\nu}- \frac 13 R_{\mu\sigma \nu\tau}(0) x^\sigma x^\tau + O(|x|^3),\\ {\Gamma^{\lambda}}_{\mu\nu}(x)= -\frac 13 (R_{\lambda\nu\mu\tau}(0)+R_{\lambda\mu\nu\tau}(0))x^\tau+ O(|x|^2). $$ Similarly we we can construct local vielbein frames in which we have a co-frame and spin connection $$ e^{*a}_\mu(x)= \delta_{a \mu}- \frac 16 R_{a \sigma \mu\tau}(0) x^\sigma x^\tau +O(x^2),\\ {\omega^a}_{b\mu}(x)=- \frac 12 {R^a}_{b\mu\tau}(0)x^\tau+O(|x|^2). $$

Answer 2

A way of constructing a local and inertial frame is using Riemann normal coordinates. As stated by S. Carroll book:

(Riemann normal coordinates) provide a realization of the locally inertial coordinates (...). They are not unique; there are an infinite number of non-Riemann normal coordinates systems (related to a neighborhood of the point p) in which $g_{\mu \nu }(p)= \eta_{\mu \nu }$ and $\partial_{\sigma} (p)=0$ but in an expansion around $p$ they will differ from Riemann normal coordinates only at third order in $x^{\mu}$.

Riemann normal coordinates lead us to the metric tensor expression:

$g_{\mu \nu }(x)= \eta_{\mu \nu} + C_{\mu \nu, \alpha \beta} x^{\alpha} x^{\beta}+.. $

where $C_{\mu \nu, \alpha \beta}$ is the second order coefficient which depends on the second derivatives of $g_{\mu \nu}$.

This definition could be taken as local if $x \approx 0$.

In this way the derivatives of Christoffel symbols have sense because, roughly speaking, the second derivative of the metric tensor (i.e. what characterizes Ch. symbols) could give a constant. In fact, as e.g. already stated here:

$\partial_{l} \Gamma^{\lambda}_{\rho \nu} = \eta^{\lambda \tau}( C_{\tau \nu, k \rho} + C_{\tau \rho , k \nu} - C_{\rho \nu , k \tau}) \delta_{l}^k + ...$

In this way we may even have:

$lim_{x \rightarrow 0}\Gamma^{\lambda}_{\rho \nu} =0$

with

$lim_{x \rightarrow 0}\Gamma^{\lambda}_{\rho \nu,\alpha} \neq 0$

In conclusion the sentence

After moving in a LIF (i.e. connections vanishes), from (4) we can isolate the derivative of Ch. symbols

isn't right because Ch. symbols are not tensors i.e. they can be zero in a frame and non-zero in another frame; for this reason the general covariance principle cannot be applied (like when we move from an the expression with ordinary derivatives to the same with covariant derivatives).