Can electron be considered a Source of gravitational field?

[big_bounce]

Hello PF members,

Source of the gravitational field in the Einstein field equations is energy–momentum tensor and the curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present.

Suppose a electron at rest .

1- Can this electron be considered as source of gravitational field ?

2- How calculate energy density, momentum density and momentum flux for a electron at rest ?

 

[bcrowell]

1- Can this electron be considered as source of gravitational field ?

Yes.

By the way, you posed the question in terms of GR, but the stress-energy tensor is equally relevant in SR. For example, there is no really satisfactory way of stating conservation of energy or equivalence of mass and energy in SR without using the stress-energy tensor.

2- How calculate energy density, momentum density and momentum flux for a electron at rest ?

It won't work if you try to describe it as a point particle. If you integrate the self-energy of the field of a point particle in classical E&M, it's infinite, whereas we know that the mass of an electron is actually finite. There are two ways you can go from here.

(1) Use quantum field theory.

(2) Don't try to model the electron as a point particle. Instead, take it to be an object whose size is at least on the order of the classical electron radius.

In option #2, you can take the expression for the Coulomb field and plug it into the expression for the stress-energy of the electric field. This will be a reasonable approximation at distances large compared to the classical electron radius.

 

[big_bounce]

Curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present.
Electron hasn't got any momentum flux or momentum density at rest. So can conclusion that curvature of space-time is directly related just to the energy ( rest energy ) for a electron at rest ?

 

[bcrowell]

How about giving us some more context on why you're posing this question in this particular way. Is there a reason that you're interested in an electron in particular? It seems that the issues I was addressing in #2 weren't really the issues you were interested in.

 

[Nugatory]

Electron hasn't got any momentum flux or momentum density at rest. So can conclusion that curvature of space-time is directly related just to the energy ( rest energy ) for a electron at rest ?

This is an I-level thread, so even if you haven't been formally introduced to tensors, it's time for you to acquire a nodding acquaintance with them. There's a pretty good introduction at http://preposterousuniverse.com/grnotes/grtinypdf.pdf .

You've hit on why the Einstein field equations use tensors. Write the stress-energy tensor using coordinates in which the electron is at rest and the elements of the tensor corresponding to momentum flux will be zero; transform to coordinates in which the electron is not at rest and these elements will take on non-zero values. However - and this is why the tensor formulation is so important and useful - the curvature scalar produced by the stress-energy tensor will come out the same either way, rather as the dot-product of two vectors comes out the same no matter what coordinates you use to write the vectors.

(There are also various tensors and vectors associated with the curvature - they also come out the same but it's less obvious because their individual components are different in different coordinate systems).

 

[Dale]

Electron hasn't got any momentum flux or momentum density at rest. So can conclusion that curvature of space-time is directly related just to the energy ( rest energy ) for a electron at rest ?

The stress energy tensor also includes stress, which is nonzero for a Coulomb field.

 

[bcrowell]

The stress energy tensor also includes stress, which is nonzero for a Coulomb field.

One way of seeing that this must be true is that if the stress-energy tensor of an electromagnetic field consisted solely of an energy density, then its trace would be nonzero. But we expect the trace to be zero because disturbances in the field propagate at c.

In any case, I doubt that we're accomplishing much without some info from the OP on why s/he is asking this particular question in this particular form.

Electron hasn't got any momentum flux or momentum density at rest. So can conclusion that curvature of space-time is directly related just to the energy ( rest energy ) for a electron at rest ?

Is there something stopping you from looking up the stress-energy tensor of the electromagnetic field and plugging in a Coulomb field for E?

 

[lightarrow]

Sorry but I haven't understood something here: which are the dimensions of the spatial region where is localized the energy of an electron "at rest"? (I'm referring to the uncertainty principle).

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