Embarassingly simple question - Christoffel symbols

[pervect]

OK, my computer program (GRTensor II) says that

$$\Gamma_{abc}$$ is symmetric in the first two indices. Which leads to the equation

$$\Gamma_{abc} = \frac{1}{2} ( \frac{\partial g_{bc}}{\partial a}+ \frac{\partial g_{ac}}{\partial b} - \frac{\partial g_{ab}}{\partial c} )$$

And that's exactly what it calculates.

Unfortunately, my textbook seems to insist that the minus sign belongs on the first term above, not the last - but there would be no way for the symbol to be symmetric in the first two indices if that were true. Another text seems to agree with the first.

Right now I'm tempted towards beliving my computer program over the textooks. This gives the result that

$$\Gamma_{xtt} = \frac{1}{2} \frac{\partial g_{tt}}{\partial x}$$

which is necessary to get the very simple result that the "forces" on a stationary body are just the gradient of g_tt in nearly flat space-time (the Newtonian limit) which Pete mentions.


Does anyone know for sure where the minus sign belongs?

 

[robphy]

Which textbook?
Did you check the sign-conventions? signature-conventions? and index-placement conventions?
[btw... I think you need, for example, "x^a" not "a" in your first expression above.]

 

[Atheist]

All textbooks I know define the Christoffel-Symbols in a way that they are symmetric in the last two indices and in that definition the minus-sign should be at the derivative by $$x^a$$ (when a is the first index as above). Also I´ve never seen any other convention anywhere else.
If your computer program uses another convention for indices I´d bet that it´s because of performace-issues (program might run faster with this method of storing data).

The physical content of both conventions is equivalent as long as other equations are altered accordingly where nessecary.

 

[pervect]

I was afraid of that. I had been assuming that my computer has been spitting out the same style results as the textbook :-(.

 

[James R]

Sounds to me like your computer is using a different definition of the symbols.

 

[robphy]

ftp://grtensor.phy.queensu.ca/pub/grtensor/doc/grCalc.ps (page c18)
says

Christoffel symbol of the first kind¹⁷
$$\Gamma_{bca} := \frac{1}{2} \left(g_{ab,c}+g_{ac,b}-g_{cb,a}\right)<br /> \hrule$$
¹⁷ Note that this index ordering differs from that of Misner-Thorne-Wheeler. We retain this ordering for consistency with earlier versions of GRTensor.

MTW (p 210), HE (p 40), ExactSolutions (p 45) say
$$\Gamma_{abc} := \frac{1}{2} \left(g_{ab,c}+g_{ac,b}-g_{bc,a}\right)<br /> \hrule$$

 

[pervect]

Thanks a lot! This clears up a lot of issues. Now I've only got the physics to wory about.