Can a Symmetric Tensor on a Manifold of Signature -+++ be Written in p-forms?

[Phrak]

Electric charge continuity is expressed as ∂_tρ + ∂_iJⁱ =0. (1)

The manifold, M in question is 3 dimensional and t is a parameter, time.
∂_iJⁱ is the inner product of the ∂ operator and J.

With M a subspace of a 4 dimensional manifold with metric signature -+++, eq. (1) can be written in forms as d*J=0, where J_μ = (J, -ρ). So electric current and charge are unified as a single vector quantity.

In other parts of physics we run into symmetric tenors. Can a symmetric tensor on a manifold of signature -+++ be written in p-forms? Or perhaps as part of a higher dimensional p-form? I'm looking for ideas...

 

[dx]

Hi Phrak,

A p-form is by definition an antisymmetric tensor. Also, what's the relevance of the signature of the metric here?

 

[Phrak]

Hello dx,

I'm interested in knowing how Einstein's stress energy tensor can be expressed in forms.

As the gradient of current density does not appear to be expressible in forms, but with the inclusion of a \partial t of \rho can be recaste as a skew symmetric tensor with lower indices, so might the stress energy tensor be expressed.

 

[waht]

Hello dx,

I'm interested in knowing how Einstein's stress energy tensor can be expressed in forms.

As the gradient of current density does not appear to be expressible in forms, but with the inclusion of a \partial t of \rho can be recaste as a skew symmetric tensor with lower indices, so might the stress energy tensor be expressed.

How about multiplying it by the metric tensor?

 

[Phrak]

Thanks, waht.

Say you have a tensor in T with metric g.

As you say, $$T_{\mu\nu} = g^{\mu} _{\sigma}g^{\nu} _{\rho}T^{\sigma\rho}$$

However, if T_uv is antisymmetric it must also have the property that $$T_{\mu\nu} = -T_{\nu\mu}$$

You spin the matrix 180 degrees around its diagonal, then also also change the sign of all the elements.

 

[Ben Niehoff]

Symmetric tensors cannot be expressed as forms, no. Unfortunately, as beautiful as forms are, they are not general enough to capture all possible kinds of linear objects. One must include tensors.

However, what you CAN do is define tensor-valued forms. If you think back to freshman electromagnetism, finding the electric field at some point by integrating along some semicircular wire or something; what you were doing was integrating a vector-valued form. So, you can simply extend that idea and get tensor-valued forms: a tensor-valued p-form is something that yields a tensor when integrated over a p-dimensional surface.

You can also have Lie-algebra-valued forms, which you can think of as matrix-valued forms. The connection form and curvature form are examples of this; they take values in the structure algebra of the manifold--for a real, Riemannian n-manifold, this is so(n).

 

[Phrak]

Thanks for your comments, Ben. I was lead into this by the equation J_uK_v = *(J/\*K), in which the direct product does not appear to be constructed of antisymmetric operations, yet could be.

Can you supply any direction in which I could prove to myself that symmetric tensors cannot be expressed as forms?

 

[Ben Niehoff]

Thanks for your comments, Ben. I was lead into this by the equation J_uK_v = *(J/\*K), in which the direct product does not appear to be constructed of antisymmetric operations, yet could be.

Where did you get that equation? It's nonsense; the free indices are not balanced.

Can you supply any direction in which I could prove to myself that symmetric tensors cannot be expressed as forms?

It's simple: because forms are always antisymmetric tensors (and the symmetry properties of any tensor are always preserved under changes in coordinates).

 

[Phrak]

That makes sense.

Where did you get that equation? It's nonsense; the free indices are not balanced.

Sorry. I can only plead exhaustion. J_μK_μ = N*(J_μ/\*K_μ), in N dimenions.