lesson 6 at last! curved manifolds · 2020. 9. 22. · review 1 a tensor eld de nes a tensor at...

Lesson 6At last!

Curved manifolds

Mario D́ıaz

The University of Texas Rio Grande Valley

[email protected]

September 22, 2020

Mario D́ıaz Curved manifolds

Mappings

A map f from a space M to a space N is a rule which associateswith an element x of M a unique element y of N. The simplestexample of a map is an ordinary real-valued function of < into

Manifolds

Examples

S2, the sphere is a manifold.

from Schutz GMofMP

A map from S2 to R2, is good for points like P, but not for Qwhich it is mapped to θ = π/2, φ = 0 and θ = π/2, φ = 2π. Andwhat about the pole?


The following is another map of S2 called the stereographic maponto R2. The map only fails at N.

from Schutz GMofMP


Other examples

1 The set of rotations of a rigid object in 3 dimensions is amanifold. (the three Euler angles provide the coordinates inR3).

2 The set of all pure-boosts Lorentz transformations (theparameters are the three components of the velocity-boost).

3 The phase space defined by the position (3N numbers) andthe velocities (3N numbers) of N particles is a manifold ofdimension 6N.

4 a vector space.

5 the set of all (x , y) solutions to a differential (or algebraic)equation for a function y(x). A particular solution will be acurve on such manifold.

6 The first one is also a Lie group which is also a C∞ manifold,i.e. SO(3).


Differential Structure

We will consider only ”differentiable manifolds”. This meansthat our manifolds will not only be continuous but also willhave maps that can be differentiated with derivatives whichwill also be well defined and differentiable except for a fewpoints.

For example:The sphere is continuous and differentiable.A cone will not be differentiable at its vertex.

Differentiability means we will be able to have vectors andone-forms and with then build all tensor types.Once we define a metric on the manifold we will have acorrespondence between forms and vectors.


Review

1 A tensor field defines a tensor at every point.

2 Vectors and one-forms are linear operators on each other,producing real numbers.

3 Tensors are also linear operators on one-forms and vectors,producing real numbers.

4 tensor operations1 multiplication by a scalar produces another tensor of the same

kind.2 addition of tensor components of the same type → produces

another tensor of the same type.3 multiplication of tensor of different types gives a new tensor of

the sum of the types, the outer product of two tensors.


4 covariant differentiation of a tensor of type

(mn

)gives a

tensor of type

(m

n + 1

).

5 Contraction on a pair of indices of the components of a tensor

of type

(mn

)gives a tensor of type

(m − 1n − 1

)(only

between and upper and lower index).

5 any tensorial equation valid in one basis, is true in any otherone.


Riemannian manifolds

Tangent space

In differential geometry, a tangent space can be defined at everypoint x of a differentiable manifold.This is a real vector space which we can pictured instinctively ascontaining the possible tangential directions that could be definedat a point x.The elements of the tangent space are called tangent vectors at x.All the tangent spaces have the same dimension, equal to thedimension of the manifold.

The tangent space to a sphere (from Wikipedia commons)Mario D́ıaz Curved manifolds

A Riemannian manifold or Riemannian space (M, g) is a realdifferentiable manifold M in which each tangent space is equippedwith an inner product given by g , a Riemannian metric, in amanner which varies smoothly from point to point. The metric gis a positive definite symmetric tensor: a metric tensor. This willlet us define angles, lengths of curves, areas (or volumes),curvature, gradients of functions and divergence of vector fields. Ina bit more rigorous but not tremendously obscure jargon aRiemannian manifold is a differentiable manifold in which thetangent space at each point is a finite-dimensional Euclideanspace. The word is honoring the German mathematician BernhardRiemann. In reality we will be using pseudo-Riemannian manifoldsbecause the metric will not be positive definite. We will be using ametric with signature (−+ ++) like the Minkowski metric. Thesemetrics will let us have in the tangent space at any point vectorswhich could have positive, negative or zero magnitude. But ourmetric will still be symmetric.


The metric and the tangent space

When Schutz talks about ”local flatness” he is describing the samenotion that we used in our definition of manifold: i.e that anypoint of the manifold will have a tangent space associated at thepoint that we associate naturally to it. The idea is that with ametric we have a unique way to associate a one-form with a vectorand vice versa. Why do we say that the metric signature has to be(−+ ++)? The reason is that we want our tangent space not tobe ”Euclidean” but ”Minkowskian”. This means that we want tohave a symmetric metric which can by linear algebra theorems bediagonalized to have eigenvalues (-1,1,1,1). The sum of thediagonal elements is called the signature. So we want our metricto have signature +2.Another important point is that in order to have coordinate basiswe need matrices Λα′β that can be turned into a Minkowskimetric. This means that we need transformations:

∂Λα′β∂xγ

=∂Λα′γ∂xβ


The local flatness theorem

We will state as Schutz’ book does, the following theorem which Iwill not prove in class. But I strongly recommend reading it.

At any point P in a pseudo riemannian manifold we can find acoordinate system{Xα} with origin at P where

gαβ(Xµ) = ηαβ + O[(x

µ)2] (1)

In an equivalent way:

gαβ(P) = ηαβ for all α, β, (2)∂

∂xγgαβ(P) = 0 for all α, β, γ; (3)

but beware:∂2

∂xγ∂xµgαβ(P) 6= 0 (4)


Lengths and ...

We recall that the metric ds2 = gαβdxαdxβ gives us the line

element ds. i.e. we get dl ≡ |gαβdxαdxβ|1/2, with which we cancalculate the length of a curve:

l =

∫alongcurve

|gαβdxαdxβ|1/2 (5)

l =

∫ λ1λ0

∣∣∣∣gαβ dxαdλ dxβdλ∣∣∣∣1/2 (6)

where λ is the parameter of the curve. We can also define a curveby the vector field tangent to it. If we have V α = dxα/dλ we get,

l =

∫ λ1λ0

|~V · ~V |1/2dλ (7)


... and Volumes

For the volume which is 4-D we have:

dx0dx1dx2dx3 =∂(x0, x1, x2, x3)

∂(x0′, x1′, x2′, x3′)dx0′dx1′dx2′dx3′, (8)

Where the indicated quotient is the Jacobian of the transformation.But if we use the local flatness theorem, all we need to see is thatt in the tangent space this Jacobian will be the product of thedeterminant of the transformation times the determinant of theMinkowski metric times the determinant of the transpose of thematrix of coordinates transformations. This result essentially saysthat the determinant is the determinant of the Minkowski metricwhich takes us to the following result, a very important one:


dx0dx1dx2dx3 = [−det(gα′β′)]1/2dx0′dx1′dx2′dx3′, (9)

= (−g)1/2dx0′dx1′dx2′dx3′. (10)

This gives the proper volume element.

Simple example

You can compare with the metric in spherical coordinates,ds2 = dr 2 + r 2dθ2 + r 2 sin2 θdφ2 If we calculate dV = dx3 incartesian it would give, remembering that the determinant isr 4 sin2 θ (watch out: the metric here is 3-D positive definite):

dV = r 2 sin θdrdθdφ (11)


Covariant Differentiation again

Differentiation of a vector in more than one dimension implies tolook at the value of a vector at given point and then after choosinga given direction (i.e. the direction along which we want tocalculate the derivative of the vector) at the value it has at adifferent point (in that given direction), located from the previousone a distance that we make infinitesimal.This infinitesimal direction would be a fuzzy concept when thespace is not flat.

That’s the reason why we need the concept of covariant derivation.Of course in a very small neighborhood of a given point when wemove around in the tangent space, the covariant derivative will bethe same as the ordinary derivative.


The divergence of a vector

V α;α = Vα,α + Γ

αµαV

µ (12)

And,

Γαµα =1

2gαβ(gβµ,α + gβα,µ − gµα,β) (13)

=1

2gαβ(gβµ,α − gµα,β) +

1

2gαβgβα,µ (14)

Now notice:

gαβgβµ,α − gαβgµα,β = gαµ,α − gµβ,β = 0 (15)

(the latter one due to the symmetry of the metric tensor!) whichmeans:

Γαµα =1

2gαβgαβ,µ (16)


Let’s review some properties of matrices: The inverse of

M =

(a bc d

)is M−1 =

1

(ad − bc)

(d −b−c a

)where (ad − bc) is the matrix determinant. The x derivative of M

is M, x =

(a, x b, xc , x d , x

). Calculating the trace of M−1M, x :

Tr(M−1M, x

)= Tr

(1

(ad − bc)

(d −b−c a

)(a, x b, xc , x d , x

))

= Tr

(1

(ad − bc)

(da,x − bc,x db,x − bd,x−ca,x + ac,x −cb,x + ad,x

))i.e., Tr

(M−1M, x

)=

(da,x +ad,x−bc,x +−cb,x )(ad−bc) =

(ad−bc),x(ad−bc) so

Tr(M−1M, x

)= (log(det(M)),x


This is a general result for a nonsingular square matrix. So we canapply it to our metric to obtain:

[log(det||gαβ||)],α = Tr(||gαβ||−1||gµν,α||)

which means:

g,αg

= gµνgµν,α and then g,α = ggµνgµν,α (17)

Using this result in (16):

Γαµα = (log(−g)1/2)),µ =(√−g),µ√−g

(18)

And then the divergence can be written:

V α;α = Vα,α + V

α (√−g),α√−g

(19)


Gauss’ law

We can take now a look at Gauss’ law where we integrate thedivergence over a volume (using the proper volume element).∫

V α;α√−gd4x =

∫(√−gV α),αd4x . (20)

And then, ∫(√−gV α),αd4x =

∮V αnα

√−gd3S . (21)

So Gauss’ law on a curved manifold is:∫V α;α

√−gd4x =

∮V αnα

√−gd3S . (22)

where ~n is a 4-vector normal to the 3-surface element d3S .


Curvature

We need to make a difference between extrinsic and intrinsiccurvature.

We define Extrinsic curvature for a space embedded in anotherspace and it is related to the radius of curvature of circles that”are” in the embedded space. For example the extrinsic curvatureof a circle is equal to the inverse of its radius everywhere.

Intrinsic curvature, instead, is defined at each point in aRiemannian manifold.An intrinsic definition of the Gaussian curvature at a point P couldbe the following: imagine the ant that Schutz describes in hisbook, which is tied to P with a short thread of length r. She runsaround P while the thread is completely stretched and measuresthe length C(r) of one complete trip around P.


If the surface were flat, she would find C (r) = 2πr .

On curved surfaces, the formula for C(r) will be different, and theGaussian curvature K at the point P can be computed by theso-called Bertrand - Diquet - Puiseux theorem as:

K = limr→0

(2πr − C (r)). 3πr 3

. (23)

When we apply this formula to a a cylinder we get 0.


A clear example of intrinsic non-zero curvature is the failure of theEuclid’s’ fifth postulate. i.e we take a look at two vectors that areparallel to each other at the equator: the curves to which they aretangent will not have its tangent vectors parallel at the north pole.


And if we try to transport one the vectors parallel to itself througha closed loop like in the figure, i.e. from A to B and then along ameridian the same way (parallel) through it until it reaches thenorth pole , and then keeping it parallel to itself again downanother meridian that passes through the point A at the equator itends up at 90◦ with itself at the origin.


Parallel transport and geodesics

The construction we made in the previous slide is called paralleltransport.When the covariant derivative of a vector field respect to itself iszero, the curves to which this vector field is tangent are calledgeodesics. It is also said that a vector field to geodesic curves isparallel transported to itself along these geodesics. For these linesthe vector tangent to the curve at one point is parallel to thevector tangent to the same curve at a previous point.

The equation that describes such curves can be obtained bywriting the equation:

∇~U ~U = 0

This can be written:

UβUα;β = UβUα,β + Γ

αµβU

µUβ = 0 (24)


If we let s be a parameter along the curve, then Uα = dxα/ds andUβ∂/∂xβ = d/ds from where we get:

d

ds

(dxα

ds

)+ Γαµβ

dxµ

ds

dxβ

ds= 0 (25)

This is called the geodesic equation.If s is a parameter along the geodesic curve, it can bere-parametrized the following way: λ = as + b.

Parameters related this way are called affine. A geodesic is a curveof extremal length between two points. This is the generalizationof the fact that the extremal (minimal) length between two pointin Euclidean space is given by a straight line. It can be shown thatthe proper distance along a geodesic is an affine parameter itselfand that is the way geodesics are typically calculated.


Intrinsic curvature and the curvature tensor

We want to parallel transport a vector ~V defined at A all throughthe closed loop A-B-C-D in the figure below.


The result naturally has to be ∇~e1 ~V = 0 which gives:

∂V α

∂x1= −Γαµ1V µ (26)

Integrating from A to B, gives:

V α(B) = V α(A) +

∫ BA

∂V α

∂x1dx1 (27)

= V α(A)−∫

x2=bΓαµ1V

µdx1 (28)

Doing the same from B to C to D:

V α(C ) = V α(B)−∫

x1=a+δaΓαµ2V

µdx2

V α(D) = V α(C ) +

∫x2=b+δb

Γαµ1Vµdx1


and back to A again,

V α(A) = V α(D) +

∫x1=a

Γαµ2Vµdx2 (29)

Adding all the equations we have the final vector:

δV α = V α(Afinal )− V α(Ainitial )

=

∫x1=a

Γαµ2Vµdx2 −

∫x1=a+δa

Γαµ2Vµdx2

+

∫x2=b+δb

Γαµ1Vµdx1 −

∫x2=b

Γαµ1Vµdx1

Notice that these terms do not cancel because Γαµν and Vµ are

not constant along the loop (we are assuming curvature).


And if we also notice that i.e. the first two terms can be written:∫x1=a

Γαµ2Vµdx2 −

∫x1=a+δa

Γαµ2Vµdx2 =∫

x1=a+δa Γαµ2V

µ −∫

x1=a Γαµ2V

µ

x1(a + δa)− x1(a)dx2 =

−∫ b+δb

bδa

∂

∂x1Γαµ2V

µdx2

we get then:

δV α ' −∫ b+δb

bδa

∂

∂x1Γαµ2V

µdx2

+

∫ a+δaa

δb∂

∂x2Γαµ1V

µdx1


which gives:

δV α ≈ δaδb[− ∂∂x1

Γαµ2Vµ +

∂

∂x2Γαµ1V

µ

](30)

Using (28) to eliminate the derivatives of V α:

δV α = δaδb [Γαµ1,2 − Γαµ2,1 + Γαν2Γ νµ1 − Γαν1Γ νµ2] V µ. (31)

Here 1 and 2 are antisymmetric because the indices refer toopposite paths. If we used general coordinate lines xσ and xλ,δV α = {change of V α} is:

1) δa→ ~eσ, 2) δb → ~eλ, 3)− δa→ ~eσ, and finally 4)− δb → ~eλ

or in better mathematical language:

δV α = δaδb [Γαµσ,λ − Γαµλ,σ + ΓανλΓ νµσ − ΓανσΓ νµλ] V µ. (32)

Between brackets we have the Riemann curvature tensor Rαβµν .


Rαβµν = Γαβν,µ − Γαβµ,ν + ΓασµΓσβν − ΓασνΓσβµ. (33)

The Riemann tensor ”measures” how much a vector transportedparallel to itself around a close loop in a given manifold, hasdeparted from being parallel. From its definition we see that if themanifold is flat the vector remains parallel to itself after thistransport.

We can look at another way of deriving it.We will use the fact that in general covariant derivation is notcommutative.We can define the commutator of covariant derivatives this way:

∇c∇d T a...b... −∇d∇c T a...b... .

Let us calculate the covariant derivative of a vector X a:

∇c X a = ∂cX a + Γ abc X b.


Then using that it is a tensor

(11

):

∇d∇c X a =∂d (∂c X

a + Γ abc Xb) + Γ aed (∂c X

e + Γ e bc Xb)− Γ e cd (∂eX a + Γ abeX b),

and,

∇c∇d X a =∂c (∂d X

a + Γ abd Xb) + Γ aec (∂d X

e + Γ e bd Xb)− Γ e dc (∂eX a + Γ abeX b),

Now we can subtract the two equations and assuming that:

∂c∂d Xa = ∂d∂cX

a

get:

∇c∇d X a −∇d∇cX a = Rabcd X b + (Γ e cd − Γ e dc )∇eX a, (34)


where we have defined Rabcd :

Rabcd = ∂cΓa

bd − ∂d Γ abc + Γ e bd Γ aec − Γ e bcΓ aed . (35)

Notice that when the metric is torsion free the parenthesis on theright hand side of (36) vanishes, so we get:

∇[c∇d ]X a =1

2Rabcd X

b. (36)

Notice that the way we define it the Riemann tensor is a

(13

)type tensor. We can describe it by saying that it ”measures” the”un-commutativity” of the covariant derivative. Now we want tocalculate R in terms of the metric (after all the metric is themeasuring element in our manifolds). We can do it in a locallyinertial frame ( a frame in which by definition the connectioncoefficients vanish and the covariant derivative is the regularderivative).


The connection coefficients at a given point P, can be computedusing the eq (13) from slide 23 in this Lesson.

Γ abc,d =1

2g ae(geb,cd + gec,bd − gbc,ed ), (37)

and then we get at P:

Rabcd =1

2g ae(geb,dc + ged ,bc − gbd ,ec

−geb,cd + gec,bd + gbc,ed ).

But we’re working with symmetric g so geb,dc = geb,cd ,and then at P:

Rabcd =1

2g ae(ged ,bc − gec,bd + gbc,ed − gbd ,ec ). (38)


We can lower the index a:

Rabcd = gaeRe

bcd =1

2(gad ,bc − gac,bd + gbc,ad − gbd ,ac ). (39)

And now it is easy to verify:

Rabcd = −Rbacd = −Rabdc = Rcdab (40)

Rabcd + Radbc + Racdb = 0. (41)

Rabcd is antisymmetric on the first and second pair of indices andsymmetric on exchange of the two pairs. Notice that (40) and (41)are true tensor equations, while (38) is not (it does not includecovariant derivatives: it is only valid in that particular frame).

Rabcd = 0⇔ flat manifold . (42)


Geodesic deviation...Or how dead is Euclides’ Fifth postulate

(The following narrative follows d’Inverno’s book) Let’s look at two geodesiccurves. They are both solutions (i.e. belong to the same family) of the

geodesic equation. λ is the affine parameter. ~V and ~V ′ are the respectivetangent vectors to each of them. We define now ~ξ which is a vector thatconnects both, i.e. from a given value of λ on one to the similar value of λ onthe other one (λ, the affine parameter could well be the proper time).


This in the same sense that (x = t, y = −5t2 + 3t) and(x = t, y = −5t2 + 3t + 2) are solutions of the geodesic equationfor a uniform potential.


We want to investigate how much do one depart form the other.Let’s look at 2-surface S spanned by what is called a congruenceof (timelike) geodesics. This is a family of geodesics such thatexactly one of the curves goes through every point of S. Theparametric equation of S is:

xa = xa(τ, λ), So then: (43)

V a =dxa

dτξa =

dxa

dλ(44)

i.e. V a is the tangent vector to the time-like geodesic at eachpoint and ξa is a connecting vector connecting two neighboringcurves in the congruence.


The commutator of V a and ξa satisfies:

[V , ξ]a = V b∂bξa − ξb∂bV a (45)

=dxb

dτ

∂

∂xb

(dxa

dλ

)− dx

b

dλ

∂

∂xb

(dxa

dτ

)(46)

=d

dτ

(dxa

dλ

)− d

dλ

(dxa

dτ

)(47)

=d2xa

dτdλ− d

2xa

dλdτ= 0 (48)

But we can (we should) replace these derivatives by a covariantderivative:

∇V ξa −∇ξV a = 0 (49)


Lie derivative

If we now take the covariant derivative of the previous equationrespect to V a (which will mean we want to see how this equationvaries along the geodesics):

∇V∇V ξa −∇V∇ξV a = 0 (50)

This suggests the following definition:For any given two vector fields, we can always define the Liebracket [~U, ~V ]:

[~U, ~V ]α = Uβ∇βV α − V β∇βUα. (51)


Notice that:

[~U, ~V ]α = UβV α;β − V βUα;β (52)

[~U, ~V ]α = UβV α,β + UβΓαδβV

δ − V βUα,β − V βΓαδβUδ (53)= UβV α,β − V βUα,β (54)

which shows that it does not depend on the metric.

This bracket is also called the Lie derivative of ~Y along ~X .

L~U ~V = [~U, ~V ] (55)


The Lie derivative can also be defined for a tensor in general:

L~X Ta...b... = X

c∂c Ta...b... − T c...b... ∂c X a + T a...c...∂bX c + · · · . (56)

Notice that the Lie derivative is linear, is Leibniz, it is ”type

preserving”, i.e. the Lie derivative of a tensor of type

(pq

)is

another tensor of type

(pq

).

Back to the study of geodesic deviation. We will use the followingresult:

∇X (∇Y Z a)−∇Y (∇X Z a)−∇[X ,Y ]Z a = Rabcd Z bX c Y d (57)

If we set X a = Z a = V a and Y a = ξa then the third termvanishes because of our previous result regarding the Lie derivativeof ξ respect toV , but ∇~V ~V = 0 so the second term also vanishes.


So we get:

∇V (∇ξV a) = Rabcd V bV cξd (58)

We can now use (53) which will gives us, valid everywhere:

∇V (∇V ξa) = Rabcd V bV cξd (59)

This means that while geodesics in a flat space-time remainparallel ( maintain their separation constant), geodesics in a curvedmanifold do not in general. The Riemman tensor is actually a”measure” of how much they don’t.


In Lesson 6 we study the situation seen when dropping twoparticles inside a lift falling down freely through a shaft near theearth. The particles are getting closer together and this is how weknow there is a gravitational field. This is a central idea leading toa theory of relativity that needs geometry in its formulation: Thepaths followed by test particles will indicate thecurvature-geometry of space.If we call geodesics these paths, howthese paths diverge or converge, i.e. how they deviate fromparallelism is what indicates the presence of a gravitational field.


The Ricci and the Einstein tensor

If we look at equation (41) we can keep working in an inertialframe and taking one more derivative get:

Rabcd ,e =1

2(gad ,bce − gac,bde + gbc,ade − gbd ,ace). (60)

Using the symmetry of gab and the commutativity of the partialderivatives:

Rabcd ,e + Rabec,d + Rabed ,c = 0 (61)

But we can generalize:

Rabcd ;e + Rabec;d + Rabed ;c = 0 (62)

which as a tensor equation is valid in any system and receives thename of: Bianchi identity.


There is only one possible contraction with the Riemann tensorthat defines a very important derived tensor, the Ricci tensor:

Rbc = Ra

bac (63)

Other contractions are possible as well,but due to the symmetriesof the Riemann tensor they will vanish or give −Rbc .

We can also contract further and define the Ricci scalar:

R = g bcRbc = gcd g abRacbd (64)

We can also explore a contraction similar to the one performed toget the Ricci tensor with the Bianchi identities.

g ac [Rabcd ;e + Rabec;d + Rabed ;c ] = 0 (65)


To do this calculation we work each term of the sum this way:

g ac Rabcd ;e = (gac Rabcd );e − g ac ;eRabcd (66)

But g ac ;e = 0 so we get: [Rbd ;e + Rbe;d + Rc

bed ;c ] = 0Contracting again with g:

g bd [Rbd ;e + Rbe;d + Rc

bed ;c ] = R;e − Rc e;c − Rc e;c = 0 (67)

Since R is a scalar

(2Rc e − δc eR);e = 0 (68)

We define now, the Einstein tensor:

G ab ≡ Rab − g abR = G ba (69)

Then the previous equation gives:

G ab ;b = 0 (70)


Summary

1 Physics will take place on Riemmanian manifolds , smoothspaces that locally resemble R4 with a metric defined on them.

2 The metric has signature +2, i.e. (−+ ++) and there alwaysexists a coordinate system in which, at a single point, we canhave

gab = ηab, (71)

gab,e = 0⇒ Γ abe = 0 (72)


3 The element of proper volume is:

|g |1/2d4x , (73)

where g is the determinant of the metric.

4 The covariant derivative is the extension of the regulardefinition of derivative accounting for curvature.

5 When the covariant derivative of a tensor along a curve is zerowe will interpret saying that the quantity has beingtransported parallel. A geodesic is a curve that transport itstangent vector parallel to itself. Its affine parameter can betaken to be the proper distance itself.


6 The Riemman tensor encodes all the information about thecurvature of the manifold. It only vanishes identically whenthe manifold is flat. It has 20 independent components, and itsatisfies the Bianchi identities, differential equations. Itdepends on the metric and its first and second partialderivatives. The Ricci scalar, the Ricci tensor, and theEinstein tensor are derived quantities constructed withcontractions of the Riemman tensor. The Einstein tensorcodifies the entire geometric content interacting with matterand energy to give the dynamics of the gravitational field, viathe Einstein’s equations:

Gµν = 8πTµν (74)

where Tµν is the energy momentum tensor. As a result of thisand (75) we have

Tµν ;ν = 0 (75)

which means that the quadri-divergence of the energymomentum tensor is a conserved quantity.


lesson 6 at last! curved manifolds · 2020. 9. 22. · review 1 a tensor eld de nes a tensor at...

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