Transformation of the Covariant Derivative

Anamitra Palit

Physicist,freelancer

P154 Motijheel Avenue,Flat C4,Kolkata 700074,India

palit.anamitra@gmail.com
Cell No:+ 919163892336

Abstract

The article considers the transformation of the covariant derivative of a rank one contravariant tensor to bring out a conflicting aspect of the theory in that we arrive at an impossible equation pointing to conflict in the theory.

Introduction

The covariant derivative of a rank one contravariant tensor is a mixed tensor of rank two. Its transformation leads to an impossible equation to bring out a contradiction in the theory.

Calculations

We consider the transformation of the covariant derivative[1] of the rank one contravariant tensor[which is a mixed tensor of rank two ]

\begin{equation}

\begin{aligned}

&\frac{\partial \bar{A}^{\mu}}{\partial \bar{x} \rho}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma}=\frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial \bar{x}^{\rho}}\left[\frac{\partial A^{\alpha}}{\partial x^{\beta}}+\Gamma_{\beta \gamma}^{\alpha} A^{\gamma}\right]\\

&\frac{\partial \bar{A}^{\mu}}{\partial \bar{x}^{\rho}} d \bar{x}^{\rho}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial \bar{x}^{\rho}}\left[\frac{\partial A^{\alpha}}{\partial x^{\beta}} d \bar{x}^{\rho}+\Gamma_{\beta \gamma}^{\alpha} A^{\gamma} d \bar{x}^{\rho}\right]\\

&d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \frac{\partial x^{\beta}}{\partial \bar{x} \rho}\left[\frac{\partial A^{\alpha}}{\partial x^{\beta}} \frac{\partial \bar{x}^{\rho}}{\partial x^{k}} d x^{k}+\Gamma_{\beta \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\rho}}{\partial x^{k}} d x^{k}\right]\\

&d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\frac{\partial x^{\beta}}{\partial \bar{x}^{\rho}} \frac{\partial \bar{x}^{\rho}}{\partial x^{k}} \frac{\partial A^{\alpha}}{\partial x^{\beta}} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}+\Gamma_{\beta \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\rho}}{\partial x^{k}} \frac{\partial x^{\beta}}{\partial \bar{x}^{\rho}} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}\\

&d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\delta^{\beta} k \frac{\partial A^{\alpha}}{\partial x^{\beta}} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}+\Gamma_{\beta \gamma}^{\alpha} A^{\gamma} \delta_{k}^{\beta} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}

\end{aligned}

\end{equation}

\begin{equation}

\begin{array}{l}

d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\frac{\partial A^{\alpha}}{\partial x^{k}} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}+\Gamma_{k \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k} \\

d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} \frac{\partial A^{\alpha}}{\partial x^{k}} d x^{k}+\Gamma_{k \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k} \\

d \bar{A}^{\mu}+\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\mu}=d \bar{A}^{\mu}+\Gamma_{k \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}

\end{array}

\end{equation}

\begin{equation}

\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=\Gamma_{k \gamma}^{\alpha} A^{\gamma} \frac{\partial \bar{x}^{\mu}}{\partial x^{\alpha}} d x^{k}

\end{equation}

-----(2)

Equation (2) holds for any manifold.In particular we consider the flat space time manifold.

In the flat space time context the Christoffel symbols[all of them] are zero only in the Cartesian system but non zero[all not zero] in the others. On the right side of equation (2)we consider Cartesian coordinates in flat space time:

\begin{equation}

\Gamma_{k \gamma}^{\alpha}=0

\end{equation}.

On the left side we consider some other coordinate system, manifold remaining the same that is flat space time.

Therefore from (2) we obtain:

\begin{equation}

\bar{\Gamma}_{\rho \sigma}^{\mu} \bar{A}^{\sigma} d \bar{x}^{\rho}=0

\end{equation}

---------------(3)

But

\begin{equation}

\bar{\Gamma}_{\rho \sigma}^{\mu} \neq 0

\end{equation}

,and the field̅

\begin{equation}

\bar{A}^{\sigma}

\end{equation}

is arbitrary! The possibility of equation (3) materializing comes into question.

We may obtain the Christoffel symbols for flat pace time in the spherical system by applying M=0 to the Schwarzschild Christoffel symbols[2]. We have six non vanishing Christoffel symbols for M=0

\begin{equation}

\Gamma_{\theta \theta}^{r}=-r, \Gamma_{\varphi \varphi}^{r}=-r \sin ^{2} \theta, \Gamma_{r \theta}^{\theta}=\frac{1}{r}, \Gamma_{\varphi \varphi}^{\theta}=-\cos \theta \sin \theta, \Gamma_{r \varphi}^{\varphi}=1 / r, \Gamma_{\theta \varphi}^{\varphi}=\cot \theta

\end{equation}

Direct Verification[flat space time, spherical]:

Following the usual technique[3],

\begin{equation}

\begin{array}{c}

\Gamma_{\beta \gamma}^{\alpha}=\frac{1}{2} g^{\alpha s}\left[\frac{\partial g_{s \beta}}{\partial x^{\gamma}}+\frac{\partial g_{s \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta \gamma}}{\partial x^{s}}\right] \\

g_{\alpha k} \Gamma_{\beta r}^{\alpha}=\frac{1}{2} g_{\alpha k} g^{\alpha s}\left[\frac{\partial g_{s \beta}}{\partial x^{\gamma}}+\frac{\partial g_{s \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta \gamma}}{\partial x^{s}}\right] \\

g_{\alpha k} \Gamma_{\beta \gamma}^{\alpha}=\frac{1}{2} \delta_{k}^{s}\left[\frac{\partial g_{s \beta}}{\partial x^{\gamma}}+\frac{\partial g_{s \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta \gamma}}{\partial x^{s}}\right] \\

g_{\alpha k} \Gamma_{\beta \gamma}^{\alpha}=\frac{1}{2}\left[\frac{\partial g_{k \beta}}{\partial x^{\gamma}}+\frac{\partial g_{k \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta \gamma}}{\partial x^{k}}\right]

\end{array}

\end{equation}

[No summation on k]

In the orthogonal system the only surviving term on the left side is

\begin{equation}

g_{k k} \Gamma_{\beta r}^{k}

\end{equation}

with no summation on k.

We have,

\begin{equation}

\begin{array}{l}

g_{k k} \Gamma_{\beta \gamma}^{k}=\frac{1}{2}\left[\frac{\partial g_{k \beta}}{\partial x^{\gamma}}+\frac{\partial g_{k \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta r}}{\partial x^{k}}\right] \\

\Gamma_{\beta \gamma}^{k}=\frac{1}{2 g_{k k}}\left[\frac{\partial g_{k \beta}}{\partial x^{\gamma}}+\frac{\partial g_{k \gamma}}{\partial x^{\beta}}-\frac{\partial g_{\beta \gamma}}{\partial x^{k}}\right]

\end{array}

\end{equation}

As for an example we may have,

\begin{equation}

\begin{array}{c}

\Gamma_{r \varphi}^{\varphi}=\frac{1}{2 g_{\varphi \varphi}}\left[\frac{\partial g_{\varphi r}}{\partial \varphi}+\frac{\partial g_{\varphi \varphi}}{\partial r}-\frac{\partial g_{r \varphi}}{\partial \varphi}\right] \\

\Gamma_{r \varphi}^{\varphi}=\frac{1}{2 g_{\varphi \varphi}} \frac{\partial g_{\varphi \varphi}}{\partial r}=\frac{1}{-2 r^{2} \sin ^{2} \theta} \frac{\partial\left(-r^{2} \sin ^{2} \theta\right)}{\partial r}=1 / r

\end{array}

\end{equation}

The other Christoffel symbols may be verified in a similar manner.[Covariant derivative reduces to partial derivative in the fat space time context only in the Cartesian system]

Conclusions

As stated at the outset we have arrived at an impossible equation starting from the transformation of the covariant derivative of a contravariant tensor. This points to difficulties in the basic theory.

References

1. Spiegel M R,Vector Analysis and an Introduction to Tensor Analysis, Schaum’s Outline Series, MacGraw Hill Book Company,Singapore,1974,Chapter 8,Tensor Analysis, problem 52,p197-198.

2. Hartle J. B.,Gravity ,Pearson Education Inc,2003,Appendix B,p570