Narit Pidokrajt www.physto.se/∼narit

Curvature calculation in a nutshell

October 5, 2003

Narit PidokrajtEmail: narit@physto.se

Abstract

I have found, from my experience as a student of General Relativity(GR) course, that the difficulties of GR do not stem only from under-standing the physics of it, but also the mathematical structures, namelydifferential geometry and tensor analysis. Therefore this article is writ-ten in order to demonstrate the reader how to do some calculations inGR vividly and explicitly. In this paper I will show in particular howthe curavature tensors and scalar are calculated in great details.

1 How to calculate the curvature of the metric

tensor

It is a very simple thing to do, you do not need to look for anything newbecause there are already certain formulas/equations to use. Suppose you aregiven the line element1 in the form

ds2 = gabdxadxb (1)

where gab is a metric tensor which is a symmetric tensor with the propertygab = gba. As an example the flat Minkowski metric in Special Relativity (SR)takes the form

ds2 = −dt2 + dx2 + dy2 + dz2 = ηabdxadxb (2)

where dxa and dxb refer to the coordinates used for the metric. The metric ηabreads

ηab =

−1 0 0 00 1 0 00 0 1 00 0 0 1

= diag(−1, 1, 1, 1) (3)

If you have already learned SR you would definitely know that this metric, ηab,is a flat metric, namely that it will lead to a zero curvature tensor and thusthe vanishing curvature scalar. But how can one show that this metric givesthe zero curvature tensor? The answer will be given here in this paper.

To obtain the Ricci curvature scalar—the curvature scalar for short, wejust have to use the following formulas step by step:

1There are many terminologies for this, for example, the invariant interval, the metricinterval, invariant distance etc.

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1. Write down the Riemann curvature tensor, its abstract form is

Rabcd = ∂dΓ

abc − ∂cΓ

abd + ΓmbcΓ

adm − ΓmbdΓ

acm (4)

where

Γabc =1

2gad(∂bgdc + ∂cgdb − ∂dgbc) (5)

and it is called a Christoffel symbol or metric connection or affinity. Notethat ∂c means ∂/∂x

c.

2. This is a step to compute the Christoffel symbols, which may look some-what abstract but you just plug in your metric and its inverse and dosome differentiations and algebraic manipulations, then it is just a pieceof cake. When you are done with the Christoffel symbol calculations,just substitute them into Eq. (4) and do everything consistently.

3. Once you have obtained the Riemann tensor, you can already see if themetric under consideration is curved or flat. This tensor is a measureof the intrinsic curvature at each point. It is flat if and only if itscomponents are zero. However you may want to contract it to obtainthe curvature scalar. And you can do so by using the following formula

Rab = Rcacb (6)

You can also obtain the Ricci tensor directly from the formula

Rab = ∂mΓmab − ∂bΓ

mam + ΓnabΓ

mmn − ΓmanΓ

nbm (7)

The Ricci tensor has a symmetry in its indices, namely

Rab = Rba (8)

Similar to the Riemann tensor, the Ricci tensor vanishes for the flatspacetime manifold.

4. Now that you have the Ricci tensor, it is just a little more effort tospend to get an eventual form of the curvature available, that is, the cur-vature scalar which is essentially an invariant curvature scalar, becauseit is independent of any position on the spacetime manifold. To get thecurvature scalar, use the formula

R = gabRab (9)

It is worth noting that it is not necessary to form the Ricci tensor fromthe Riemann tensor and contract it to obtain the curvature scalar. It ispossible to contract the Riemann curvature tensor down to the curvaturescalar by

R = gabRcacb (10)

So now if you follow these steps, you will definitely be able to calculate thecurvature tensor and scalar for any metric in question. Always be consistentwith your notations such as positions of the indices.

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2 But how do I calculate the curvature of the

metric?

Didn’t I already give you methods for doing it? Hmmm...anyway I know thatsometimes formulas are not enough, you would understand things better bylearning from examples. Thus I will show you one good example, namely the2-sphere’s curvature tensors and scalar. I will try to do it as explicitly as Ipossibly can.

The 2-sphere can be viewed as a spherical surface, once you are on it youwill not be able to tell whether it is a curved surface if you only stand on aparticular point. However if you start to move and make some measurement,you may be able to see how curved or flat it is. In differential geometrylanguage we do it by the so-called parallel transport method. Nevertheless theabove sentences will still not tell you how to really do the calculation. I willcome to the point by stating that the metric for the 2-sphere is given by2

ds2 = r2dθ2 + r2 sin2 θ dφ2 (11)

where θ and φ are conventional coordinates used in the spherical coordinatessystem. Note that r is just a constant in your viewpoint—as a person stand-

ing on the 2-sphere you do not see a sphere . Only those residing in the3 dimensional world would perceive r as a radius of the sphere. Recall that wecan write the metric in Eq. (11) in the same way as that in Eq. (1), so yourmetric gab will just be

gab =

(

r2 00 r2 sin2 θ

)

(12)

Writing its components, we have g11 = r2 and g22 = r2 sin2 θ. Because we willneed to employ formulas in Eq. (4) and Eq. (5), so we compute the inverse ofthe above metric, that is

gab =

(

1r2

00 1

r2 sin2 θ

)

(13)

If you happen not to remember how to inverse the metric, the simple formulais just

gij =cofactor of gij

determinant of gij(14)

Note also that gabgab = 2!—it is just a contraction of the 2 metric tensors.And now you are in a position to compute the Christoffel symbols using Eq.(5)—what you need are just the given metric and its inverse that we havewritten down above. So I will now show you some explicit calculations:

Γ111 =

1

2g1d(∂dgd1 + ∂1gd1 − ∂dg11) = 0 (15)

2This metric given to you is equivalent to a meter stick that you can use to measure thedistance on the 2-sphere you stand on.

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Why so? It is because the sum over d is limited only to d = 1, there are nooff-diagonal terms in gad. This applies to any diagonal metric! So if you areconfronted with Γ1

11, just relax—there is nothing worry. Next is,

Γ112 = Γ1

21 =1

2g1d(∂1gd2 + ∂2gd1 − ∂dg12) = 0 (16)

Again, there is no term that gives non-zero quantity. Unclear? You maythink that the for d = 1 some term will survive, but sorry...even ∂2g11 = 0 ⇔∂r2/∂θ = 0! It is explicit now? Notice the symmetry in the lower indices ofthe Christoffel symbols in Eq. (16).

Now comes something a bit more interesting,

Γ122 =

1

2g1d(∂2g2d + ∂2g2d − ∂dg22)

= − sin θ cos θ ⇔ it is nonzero for d=1! (17)

Other terms—which you should by now be able to compute by yourself—arethe following:

Γ211 = 0

Γ212 = Γ2

21 = cot θ

Γ222 = 0 (18)

Thus we have 2 non-vanishing Christoffel symbols for the metric in Eq. (12)

Γ122 = − sin θ cos θ

Γ212 = Γ2

21 = cot θ (19)

and we will use them for obtaining the Riemann tensor. Using formula in Eq.(4) what you will do is similar to what we did above to obtain the Christoffelsymbols. I will show the results as follows:

R1112 = ∂2Γ

111 − ∂1Γ

112 + Γm11Γ

12m − Γm12Γ

11m = 0 (20)

where as

R1212 = −R1

221 = ∂2Γ121 − ∂1Γ

122 + Γm21Γ

12m − Γm22Γ

11m = − sin2 θ (21)

If it is not clear to you why the above equation is so, it is because for m = 1 itvanishes , only for m = 2 and the rest is just consistent manipulation. Anotherterm that is non-vanishing is

R2112 = ∂2Γ

211 − ∂1Γ

212 + Γm11Γ

22m − Γm12Γ

21m = 1 (22)

You will surely find out the same result, just work it out carefully. And believeit or not that the remaining terms are just zero!

R2121 = R2

212 = R2221 = Ri

j11 = Rij22 = 0 (23)

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Up to now we have only two remaining components of the Riemann tensor. AsI said early on the (spacetime) metric is flat if and only if its Riemann tensoris a zero tensor, namely its components are zero. So we do have here a curvedspacetime or manifold. And we are now in a good shape to make things alot simpler ⇒ we will contract the Riemann tensor! We contract it with thecontravariant index and the last covariant index,

Rab = Rmabm =

(

R1111 +R2

112 R1121 +R2

122

R1211 +R2

212 R1221 +R2

222

)

(24)

Plugging in the Riemann tensor components we have obtained above into Eq.(24), yields the Ricci tensor!

Rab =

(

1 00 sin2 θ

)

(25)

Still your curvature tensor, though in a very simple form, is not free of coor-dinates. To obtain the curvature scalar, we contract it with gab as follows:

R = gabRab = gθθRθθ + gφφRφφ =1

r2+

sin2 θ

r2 sin2 θ=

2

r2(26)

The curvature scalar above tells you that: the larger the radius, the closer thecurvature approaches zero, or the flatter the local surface will be at each point.After all this is a result of parallel transport method!

Finally if you think you have understood this procedure and you wish tosave some time and prefer to work with a single simple formula to obtain thecurvature scalar, given the metric. Here is the thing:

Suppose you are given the diagonal metric

ds2 = gabdxadxb ⇒ gab =

(

g11 00 g22

)

(27)

The formula for the curvature scalar will be

R =1√g

[

∂1

(

1√g∂1g22

)

+ ∂2

(

1√g∂2g11

)]

(28)

where√g =

√detg. And this’s it, folks!

Have a good time!

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