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Carlo Beenakker
Carlo Beenakker
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• Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

Addendum: It seems Einstein was also not quite consistent with respect to the minus sign; in a 1914 paper he defined the $\Gamma$ without the minus sign:

In an interesting discussion on HSM it is suggested Einstein chose the symbol $\Gamma$ to refer to the first letter of "Gravitation".

• Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

• Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

Addendum: It seems Einstein was also not quite consistent with respect to the minus sign; in a 1914 paper he defined the $\Gamma$ without the minus sign:

In an interesting discussion on HSM it is suggested Einstein chose the symbol $\Gamma$ to refer to the first letter of "Gravitation".

added 425 characters in body
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.1k
  • 18
  • 448
  • 651

• Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.

• Why was the minus sign dropped?
It was understood that the identification of the Christoffel symbol with the gravitational field is mistaken: you can have a nonzero $\Gamma^\sigma_{\mu\nu}$ and zero gravitational field, all you have to do is to introduce curved coordinate systems in flat space. And conversely, the Christoffel symbol can vanish along a geodesic even if the gravitational field is nonzero.

Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.1k
  • 18
  • 448
  • 651

Why Einstein introduced a minus sign in the definition of the second Christoffel symbol $\Gamma^\sigma_{\mu\nu}$:
He writes just below equation (45) in Ref. 1:

So he wanted to identify the $\Gamma^\sigma_{\mu\nu}$ with the components of the gravitational field, and for that identification the minus sign is needed.