Other forms of the Bianchi identity

Other forms of the Bianchi Identity Sumanto Chanda S.N. Bose National Centre for Basic Sciences  JD Block, Sector-3, Salt Lake, Calcutta-700098, INDIA.

March 25, 2014

1

Intr Introd oduc ucti tion on : n-for n-forms ms

Very often the same mathematical law, takes various forms when converted to other representations. For students who start by studying these converted forms, it is sometimes difficult, and one has to be lucky to notice the similarities. We learn of the Bianchi identity when we study curvature, but a further exploration reveals otherwise. otherwise. There There are identitie identitiess we learn in vector analysis analysis that can be considered considered another another form of  the same property. This article reveals how in extensive detail. We start by first considering considering the theory of n-forms. n-forms. Suppose that we have have an object ωµn1 µ2 ...µn with n indices indices such that it exhibits exhibits anti-symme anti-symmetry try under permutation permutation of any pair of indices. We can therefore define: ω n =  ω µn1µ2...µn dxµ1 ⊗ dxµ2 ⊗ ... ⊗ dxµn n

=  ω µn1µ2...µn



dxµi

where

ωµn1...µi...µj...µn  = −ωµn1...µj...µi...µn

(1.1)

i=1

Now, we can see that for a 2-form applying (1.1) shows us that: 1 2 2 2 ω 2 =  ω µν  dxµ ⊗ dxν  = ωµν dxµ ⊗ dxν  + ωνµ dxν  ⊗ dxµ 2 1 2 2 dxµ ⊗ dxν  − ωµν  dxν  ⊗ dxµ = ωµν  2 1 2 1 2 = ωµν  dxµ ⊗ dxν  − dxν  ⊗ dxµ  = ωµν  dxµ ∧ dxν  2 2 1 2 2 ∴ (1.2) ω 2 =  ω µν  dxµ ⊗ dxν  = ωµν  dxµ ∧ dxν  2 Showing clearly that we have transmitted the anti-symmetry property of the object to the tensor product. For a n-form, we can similarly use the anti-symmetry property to show:

 

 p



 



n

1 ωn = ωµn1...µn dxµi dxµj  p! i=1  j = p+1

        p

1 1 = ωµn1...µn (−1) p−m  p!  p + 1 =0 m  p+1

 p

m

dx

i=1

µi

⊗ dx

µp+1

⊗

   dx

 j =m+1

n

1 ωµn1...µn dxµi dxµj = ( p + 1)! i=1  j = p+2

1

n

µj

k= p+2

dxµk

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2

EXTERIOR EXTERIOR DERIV DERIVA ATIVES OF N-FORMS  N-FORMS 

 p



where



 p−m

(−1)

m=0

 p+1

 p

m

dx

µi

⊗ dx

µp+1

i=1

⊗

   dx

µj

 =

 j =m+1

dxµi

(1.3)

i=1

Thus, we can repeat the process through all iterations necessary, so that we can finally write: n

1 ω = ωµn1µ2...µn dxµi n! i=1



n

(1.4)

Now we turn our attention to the exterior derivative.

2

Exter Exterior ior deriv derivati ativ ves of n-form n-formss

We start by looking at the exterior derivative of a 1-form, described by: δω 1 =  δ (ωµ1 dxµ ) =  δ (ωµ1 )dxµ + ωµ1 δ (dxµ )

=  ∂ ν ν ωµ1 δx ν  ⊗ dxµ + ωµ1 d(δx µ ) =  ∂ ν ν ωµ1 δx ν  ⊗ dxµ − d(ωµ1 )δx µ + d(ωµ1 δx µ ) =  ∂ ν ν ωµ1 δx ν  ⊗ dxµ − ∂ ν ν ωµ1 δx µ ⊗ dxν  + d(ωµ1 δx µ )





=  ∂ ν ν ωµ1 δx ν  ⊗ dxµ − δx µ ⊗ dxν   + d(ωµ1 δx µ ) The rule followed here is that the derivative performed later must appear to the left in the tensor product series, ie.:- δ  always   always appears left of  d. Writing as integrals between limits, we can say:

  

                  

2

δ 

1

2

ωµ1 dxµ  =

1

         

b

b

1 ν  µ µ ν  +d ∂ ν  ν ωµ δx ⊗ dx − δx ⊗ dx

a

a

dω1

≡

V  

1 ν  µ µ ν  ∂ ν  + ν ωµ δx ⊗ dx − δx ⊗ dx dω1

dω 1 +

=

V  

ωµ1 δx µ

∂V  



 

ωµ1 δx µ

(2.2)

 

ω

(2.1)

(2.3)

∂V  

where V  spans the region  δ ⊗ d  and  ∂ V  is the boundary of  V   V   spanning  δ . We can also see Stoke’s Theorem hidden within (2.3) as:

  

2

δ 

1

       

ωµ1 dxµ  =

2

1

b

a



  

b

1 1 ν  µ ∂ ν  ν ωµ − ∂ µ ων  δx ⊗ dx +d



dω 1

 

a

ωµ1 δx µ



 

(2.4)

Henceforth, writing δx µ ≡ dxµ we can write the volume integral part of (2.2) as one of the following equivalent forms:

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3

KNOWN KNOWN FORMS FORMS IN VECTOR VECTOR ANAL ANALYSI YSIS  S 

Thus, the exterior derivative of an n-form can be written as:

  

  

n

n

1 1 1 ν  µi  = dω = ∂ ν  ω dx dx ∂ [ν ωµ1 1...µn] dxν  ∧ dxµi ∧ ν  µ1...µn n! (n + 1)! i=1 i=1 n

2.1 2.1

 

(2.8)

For flat flat spac spaces es

Now, we consider the 2nd order exterior derivative of an n-form ω n, we can write using (2.8) : n d2 ω n =  d (dω n ) =  dx λ ∂ λ {dxν ∂ ν  ν ω }

=  dx λ ∧ dxν  ∧ ∂ λ ∂ ν ν ωn 1 = (dxλ ∧ dxν ) ∧ ∂ λ ∂ ν ν ω n + (dxν  ∧ dxλ ) ∧ ∂ ν ν ∂ λ ω n 2 1 = (dxλ ∧ dxν ) ∧ ∂ λ ∂ ν ν  − ∂ ν ν ∂ λ ω n =  0 2







d2 ω n = 0

∴

 (2.9)

This is due to the property  ∂ µ ∂ ν ν  =  ∂ ν ν ∂ µ . Such a property is easily valid on flat manifolds where the derivative operators commute as partial derivatives. It will also be valid on curved spaces where we employ covariant derivatives as we shall see.

2.2

For curv curved ed spac spaces es

In the case of curved spaces we use covariant derivatives, where ∇a ∇b  = ∇b ∇a . Howeve However, r, we are not just talking about the commuting property of the concerned derivative operators, but about the resulting n + 2 forms. For this purpose, we must remember that for n-forms, the following holds: n

ω

n

=  ω µn1µ2 ....µn



n

dx

µi

=

˜ an1 a2 ....an ω

i=1

˜ an1 a2 ....an  is ω



dy ai

(2.10)

i=1 n of  ω µ1µ2....µn ,

where, the equivalent flat-space counterpart related via diffeomorphism. Thus, we can apply (2.10) for exterior derivatives too since they are also symplectic forms. ie.:˜n Dω n =  d ω D2 ω n =  d 2 ω ˜n = 0

(2.11) (2.12)

This will event eventually ually allow us to establish the Bianchi Bianchi identit identity y on curved spaces. The  Darboux Theorem  states that at least locally such diffeomorphic transformations are possible.

3

Kno Known form formss in vecto ector r ana analy lysi siss

While equation (2.2) and (2.3) may appear unfamiliar, most students will already have encountered it’s form in vector analysis in Euclidean spaces:

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4

THE THE BIA BIANC NCHI HI IDEN IDENTIT TITY  Y 

On a n-dimensional space, we can write (3.1) and (3.2) in tensor form in a manner of (2.5) as: εµ1 µ2 ...µn ∇µ1 ∇µ2 φ  = 0

εµ1 µ2 ...µn ∇µ1 ∇µ2 (gµ3ν V  ν ) = 0

(3.3)

We will now proceed to examine more complex versions of these familiar equations.

4

The The Bian Bianc chi iden identi titty

When dealing with electromagnetic field tensor in Minkowski space-time, upon re-invoking (2.6), the Bianchi identity is given by: A  =  A ν dxν 

(4.1)

dA  =  ∂ µ Aν  dxµ ∧ dxν 

=

1 1 µ ν  µ ν  ∂ µ Aν  − ∂ ν  F µν  ν Aµ dx ∧ dx = µν  dx ∧ dx =  F  2 2





1 µ ν  ∂ µ Aν  − ∂ ν  ν Aµ dx ∧ dx 2 Invoking (2.8) and (2.9), we can therefore write: ∴

F  =  dA  =





(4.2)

dF  =  d 2 A  = 0

1 λ µ ν  ∂ λ F µν  µν  dx ∧ dx ∧ dx = 0 2

1 λ µ ν  µ ν  λ ν  λ µ ∂ λ F µν   = 0 µν  dx ∧ dx ∧ dx + ∂ µ F νλ νλ dx ∧ dx ∧ dx + ∂ ν  ν F λµ λµ dx ∧ dx ∧ dx 3! 1 λ µ ν  ∂ λ F µν  µν  +  ∂ µ F νλ νλ  +  ∂ ν  ν F λµ λµ dx ∧ dx ∧ dx = 0 3!





∴



d2 A  = 0

⇒

∂ λ F µν  µν  +  ∂ µ F νλ νλ  +  ∂ ν  ν F λµ λµ = 0



(4.3)

But this identity is also known in Riemannian geometry in the study of the curvature tensor. We know the formula for usage of the curvature tensor to be given by the exterior derivative of a 1-form in curved space by: Dω 1 = ∇µ ων 1 dxµ ∧ dxν  D2 ω1 = ∇µ ∇ν ωλ1 dxµ ∧ dxν  ∧ dxλ

1 ∇µ ∇ν  − ∇ν ∇µ ωλ1 dxµ ∧ dxν  ∧ dxλ 2 1 1 = ωρ Rµνλ ρ dxµ ∧ dxν  ∧ dxλ 2 Therefore, in accordance with (2.12), we have: =





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5

THE THE JAC JACOB OBII IDE IDENT NTIT ITY  Y 

Rµνλ ρ + Rνλµ ρ + Rλµν ρ = 0

∴

(4.4)

Now we look at the other form of the Bianchi Bianchi identity identity with the Riemann Riemann tensor. For this, we continue from the last steps that gave us (4.4) as shown below:

        

D3 ω1 =  D D2 ω1  = 0

⇒

1 D ωρ1 Rµνλ ρ dxµ ∧ dxν  ∧ dxλ = 0 2 ∇γ  ωρ1Rµνλ ρ dxγ  ∧ dxµ ∧ dxν  ∧ dxλ = 0

⇒

∇γ ωρ1 dxγ  ∧ Rµνλ ρ dxµ ∧ dxν  ∧ dxλ +ωρ1 ∇γ Rµνλ ρ dxγ  ∧ dxµ ∧ dxν  ∧ dxλ = 0

⇒

 

0

∇γ Rµνλ ρ dxγ  ∧ dxµ ∧ dxν  ∧ dxλ = 0

⇒

Thus, as before for (4.4), we get:







∇γ Rµνλ ρ + ∇µ Rνλγ ρ + ∇ν Rλγµ ρ dxγ  ∧ dxµ ∧ dxν  ∧ dxλ = 0

(4.5)

which is very much similar in appearance to the form obtained in (4.3).

5

The The Jaco Jacobi bi iden identi titty

The Jacobi identity identity can be seen as another another version of the Bianchi Bianchi identity identity. Here, Here, the indices are cycled cycled around around in the form form of a set of scalar scalar or vector vector fields. fields. It can also also be seen seen to arise from the same rule of (2.9), as shown below. We begin by considering the wedge product of the first order exterior derivatives of any two arbitrary functions, and then we consider their mixed second order exterior derivative:

 

 

df  ∧  ∧ dg  =  ∂ µ f ∂ ν ν g dxµ ∧ dxν 

=  ∂ µ f ∂ ν ν g dxµ ⊗ dxν  − dxν  ⊗ dxµ

        

= ∂ µ f ∂ ν ν g − ∂ ν ν f ∂ µ g dxµ ⊗ dxν 

≡ f, g

   

df  ∧  ∧ d dg ∧ dh  = −d df  ∧  ∧ dg ∧ dh



 = 0

= −∂ µ ∂ ν ν f ∂ ρ g ∂ σ h dxµ ∧ dxν  ∧ dxρ ∧ dxσ

1 = − ∂ µ ∂ ν ν f ∂ ρ g ∂ σ h dxµ ∧ dxν  ∧ dxρ ∧ dxσ + dxρ ∧ dxσ ∧ dxν  + dxσ ∧ dxν  ∧ dxρ 3 1 = − ∂ µ ∂ ν ν f ∂ ρ g ∂ σ h+∂ ν ν g ∂ ρ h ∂ σ f  + ∂ ν ν h ∂ ρ f ∂ σ g dxµ ∧ dxν  ∧ dxρ ∧ dxσ 3 1









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6

CONC CONCLU LUSI SION  ON 

Now, given a Poisson structure θ = 12 θµν ∂ µ  ∧ ∂ ν ν    defined on the manifold, we can define the Poisson Bracket on the manifold as:

           f, g

θ

 =  θ (df  ∧  ∧ dg ) =  θ (df, dg)

(5.2)

So (5.1) for a specific Poisson structure becomes: f, g, h

∴

θ

 + g, h, f 

θ

 + h, f, g

=0

θ

(5.3)

Thus, we can see that the Jacobi identity is connected to the identity (2.9). If using the Poisson structure θ  we can define vector fields connected to the functions f , g and h, as shown below:

 

V f  f  =  θ df, ·

V f  µ = −θµν ∂ ν  ν f 

,

 

(5.4)

then the Poisson Brackets can be described as:

                                                                    V g , V h  =  V g V h − V h V g = g, h, ·

= − h, ·, g

⇒

V g , V h  =

g, h

V f  f , V g , V h  = f,

=−

⇒

V f  f , V g , V h  =

θ

,·

g, h

g, h

θ

f, g, h

θ

θ

θ

θ

− ·, g, h

θ

− h, g, · θ

θ

θ

− h, g, ·

θ

 

θ

θ

θ

−

, ·, f 

θ

θ

,·

θ

θ

θ

,·

θ θ

, f, ·

θ

θ

− ·, f, g, h

θ

θ

g, h

θ

θ

−

g, h

θ

, f, ·

 

θ

(5.5)

θ

(5.6)

Therefore, according to (5.6) for a specific Poisson structure, (5.3) becomes: V f  f , V g , V h

6

B B

 + V g , V h , V f f 

B B

 + V h , V f f , V g

B B

=0

(5.7)

Conclus clusio ion n

Thus, we have seen that (4.6) has a similar form to the result of (4.1) and on looking closely, (3.2), the propert property y of exactnes exactness. s. The ident identit ity y in (4.5) is analogo analogous us to (3.1). (3.1). This This can be confirm confirmed ed on looking looking at (3.3). (3.3). In fact, we could could say say that that (3.1) (3.1) and (3.2) (3.2) differ from from (4.5) (4.5) and (4.6) (4.6) only in the sense that the former start from scalar functions and proceed to it’s covariant derivative, while the latter start with a vector and proceed to it’s covariant derivative. In tabular form, we could write them as: Proper perty

Vector Analysis

Riemannian Geome ometry

Irrotational

  × ∇   φ  = 0 ∇

Rµνλ γ  + Rνλµ γ  + Rλµν γ  = 0

Solenoidal

  ∇   × V    ∇

0

∇ R

γ 

+∇ R

γ 

+∇ R

γ 

0