Is the sign of the Einstein equation right in Chandrasekhar's Black holes?

[timeant]

In Chandrasekhar's book, The Mathematical Theory of Black Holes.
The sign of Einstein equations is minus "-" , Eq. (1-236).
However, the sign of Riemann and Ricci tensor are the same as MTW's book.
The sign of Einstein equations in MTW's book are "+"!

Is there a error?

 

[vanhees71]

I'd have to check this in more detail, but you have a lot of sign conventions with the metric, the curvature tensor and the Ricci tensor. For sure Chandrasekhar has the west-coast metric (+---) and if I remember right MTW have the east-coast metric (-+++). Then another sign difference can come from the definition of the curvature tensor and over which index pair of it you contract to get the Ricci tensor.

 

[PeterDonis]

Is there a error?

No, just a different choice of sign conventions. If you have a copy of MTW, look at the page titled "SIGN CONVENTIONS", just after the copyright page. You will see three different possible choices of sign convention described. On the next page is a table of the sign conventions used in various textbooks published up to the time MTW was published in 1973. Chandrasekhar's book was published later so it isn't listed in the table, but you should be able to figure out which conventions he is using from the information given in MTW.

 

[timeant]

I'd have to check this in more detail, but you have a lot of sign conventions with the metric, the curvature tensor and the Ricci tensor. For sure Chandrasekhar has the west-coast metric (+---) and if I remember right MTW have the east-coast metric (-+++). Then another sign difference can come from the definition of the curvature tensor and over which index pair of it you contract to get the Ricci tensor.

However, metric sign do not affect the sign of Ricci tensor, (1,3) Riemann tensor and Einstein equation. I think so.

Chandrasekhar use the metric(+---) for Einstein equations. However, his electromagnetic stress-energy tensor T_{\mu\nu} is base on the the metric(-+++). May be I make mistakes?

 

[PeterDonis]

Chandrasekhar use the metric(+---) for Einstein equations. However, his electromagnetic stress-energy tensor T_{\mu\nu} is base on the the metric(-+++). May be I make mistakes?

The sign convention for the metric tensor is one of only three sign conventions involved. See the page in MTW that I referenced.

 

[timeant]

Chandrasekhar's book is the same as Landau's book(The Classical Theory of Fields, 4th edn) except for the sign of Einstein equations.
Is Landau wrong? I do not think so.

In another words, the time-time component of stress-energy tensor T_{ab} is negative, i.e. T_{00} <0, in Chandrasekhar's book.
It was an unusual choice!

 

[PeterDonis]

Chandrasekhar's book is the same as Landau's book(The Classical Theory of Fields, 4th edn) except for the sign of Einstein equations.

Yes, so that just means Chandrasekhar has made the opposite choice for Landau for one of the three sign conventions that MTW describes.

Is Landau wrong? I do not think so.

Why would Landau have to be wrong? Sign conventions are conventions. You can make either choice.

 

[vanhees71]

Let's see. I've Landau-Lifshitz vol. 2, fourth edition in the reprinted version of 1996 (English version) as well as the corresponding German version, which are the same concerning the sign conventions in GR. Note that earlier editions of this book had different sign conventions (see the table in MTW @PeterDonis mentioned above).

Landau Lifshitz (4th edition)

The signature of the metric is west-coast: $\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$ for flat spacetime wrt. a Lorentzian basis.

The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=\frac{8 \pi G}{c^4} T_{\mu \nu}.$$

Chandrasekhar, The mathematical theory of black holes (1983)

The signature of the metric is west-coast: $\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$ for flat spacetime wrt. a Lorentzian basis.

The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=-A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=-\frac{8 \pi G}{c^4} T_{\mu \nu}.$$
So the difference in sign is due to the definition of the Riemann curvature tensor (note that the Riemann tensor is antisymmtric wrt. to the 1st and 2nd as well the 3rd and 4th index), and the EFEs are the same in both books.

I'd be very surprised, if there were different sign conventions for the energy-momentum tensor of matter and radiation, because usually one defines energy densities as positive, $T^{00}>0$.

 

[timeant]

Let's see. I've Landau-Lifshitz vol. 2, fourth edition in the reprinted version of 1996 (English version) as well as the corresponding German version, which are the same concerning the sign conventions in GR. Note that earlier editions of this book had different sign conventions (see the table in MTW @PeterDonis mentioned above).

Landau Lifshitz (4th edition)

The signature of the metric is west-coast: $\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$ for flat spacetime wrt. a Lorentzian basis.

The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=\frac{8 \pi G}{c^4} T_{\mu \nu}.$$

Chandrasekhar, The mathematical theory of black holes (1983)

The signature of the metric is west-coast: $\eta_{\mu \nu}=\mathrm{diag}(1,-1,-1,-1)$ for flat spacetime wrt. a Lorentzian basis.

The Riemann curvature tensor is defined by
$$\nabla_{\mu} \nabla_{\nu} A^{\rho}-\nabla_{\nu} \nabla_{\mu} A^{\rho}=-A^{\sigma} {R^{\rho}}_{\sigma \mu \nu}$$
and the Ricci tensor by
$$R_{\mu \nu} = g^{\rho \sigma} R_{\rho \mu \sigma \nu}.$$
The Einstein field equations read
$$R_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}=-\frac{8 \pi G}{c^4} T_{\mu \nu}.$$
So the difference in sign is due to the definition of the Riemann curvature tensor (note that the Riemann tensor is antisymmtric wrt. to the 1st and 2nd as well the 3rd and 4th index), and the EFEs are the same in both books.

I'd be very surprised, if there were different sign conventions for the energy-momentum tensor of matter and radiation, because usually one defines energy densities as positive, $T^{00}>0$.

I do not find the above definitions in two books.

I will give the component equations.
Chandrasekhar, The mathematical theory of black holes (1983)
The Riemann curvature tensor is defined by Eq 1-154
$$R^j_{\cdot lnm}= \Gamma^j_{lm,n}- \Gamma^j_{ln,m}+ \Gamma^j_{pn}\Gamma^p_{lm}-\Gamma^j_{pm}\Gamma^p_{ln}$$
The Ricci curvature tensor is defined by Eq 1-172
$$R_{lm}=R^n_{\cdot lnm}= \Gamma^n_{lm,n}- \Gamma^n_{ln,m}+ \Gamma^n_{pn}\Gamma^p_{lm}-\Gamma^n_{pm}\Gamma^p_{ln}$$

Landau Lifshitz (4th edition)
The Riemann curvature tensor is defined by Eq 91.4
$$R^i_{\cdot klm}= \Gamma^i_{km,l}- \Gamma^i_{kl,m}+ \Gamma^i_{pl}\Gamma^p_{km}-\Gamma^i_{pm}\Gamma^p_{kl}$$
The Ricci curvature tensor is defined by Eq 92.7
$$R_{km}=R^l_{\cdot klm}= \Gamma^l_{km,l}- \Gamma^l_{kl,m}+ \Gamma^l_{pl}\Gamma^p_{km}-\Gamma^l_{pm}\Gamma^p_{kl}$$

Are they different?
I think they are the same. right?

 

[vanhees71]

I translated the definitions in the textbooks to my preferred notation. I don't like the "comma" or "semicolon" notation for derivatives and covariant derivatives.

For Chandrasekhar I inferred it from Eq. (231) and made use of the antisymmetry of the Riemann tensor in interchanging the first two components. For Landau Lifshitz I used Eq. (91.7) and made use of the antisymmetry in interchanging the 3rd and 4th index.

 

[timeant]

It is very clear that $T_{ab}=-e U_a U_b$ for perfect fluid in Chandrasekhar's book.
Otherwise, it will be different from Landau's book.

 

[vanhees71]

Strange. Comparing Chandra's (231) with the corresponding formula in Landau-Lifshitz, I think the sign is due to the sign definition of the Riemann and Ricci tensors. On the other hand, judging from the expressions (334) for the electromagnetic-field EM tensor you are right. He has it with the opposite sign, which is indeed very strange. I'm now as puzzled as you are :-(.

 

[timeant]

Strange. Comparing Chandra's (231) with the corresponding formula in Landau-Lifshitz, I think the sign is due to the sign definition of the Riemann and Ricci tensors. On the other hand, judging from the expressions (334) for the electromagnetic-field EM tensor you are right. He has it with the opposite sign, which is indeed very strange. I'm now as puzzled as you are :-(.

No puzzles!
MTW said: All authors agree positive energy density, $T_{00}>0$!
Chandrasekhar said: No, I am an exception!

All equations are correct in Chandrasekhar's book. It's just that the energy density is negative.

 

[vanhees71]

Then I must have made a mistake in reading off the sign convention for the Riemann tensor in LL or Chandrasekhar's book. Let me check again:

Landau: (91.6)
$$A_{i;k;l}-A_{i;l;k}={R^m}_{ikl} A_m=\nabla_l \nabla k A_i-\nabla_k \nabla_l A_i.$$
Now I see that indeed Chandrasekhar (231) is indeed the same. Somehow I was obviously confused from Eq. (91.7) in LL (having overlooked that there the contraction was wrt. the second rather than the first index, while in (91.6) the contraction is over the first index of the Riemann tensor).

Sorry for that blunder :-(((!

But to make energies negative is really very unconventional. What a confusion!