Real relativistic field and curvature?

[Spinnor]

Say we have a real field that satisfies:

E^2 = P^2 + m^2

Assume spacetime is 4D. Assume the field is at rest and grab a single point of this field and slowly displace it a distance x. Just as an anchored string (string with an additional sideways restoring force) with fixed end points will have its length change when a point is displaced and just as a two dimensional "anchored" membrane will change its area when a single point is displaced can we say that a 3 dimensional "anchored" membrane will change its volume if a single point is displaced a distance x? When I say "anchored" membrane it is the real relativistic field I am thinking of.

For small displacements, x, the change in volume is proportional to what power of x?

Thanks for any suggestions on how to solve this.

 

[Spinnor]

I'm thinking the change in volume will also depend on the radius of volume around the point that is displaced?

Thanks for any help.

 

[Spinnor]

And then we would like to let the point go and picture volume changing with time and space. But first things first.

 

[Spinnor]

And then repeat all the above questions for a massless real field,

E^2 = P^2

 

[Spinnor]

Say we have a real field that satisfies:

E^2 = P^2 + m^2

Assume spacetime is 4D. Assume the field is at rest and grab a single point of this field and slowly displace it a distance x. Just as an anchored string (string with an additional sideways restoring force) with fixed end points will have its length change when a point is displaced and just as a two dimensional "anchored" membrane will change its area when a single point is displaced can we say that a 3 dimensional "anchored" membrane will change its volume if a single point is displaced a distance x? When I say "anchored" membrane it is the real relativistic field I am thinking of.

For small displacements, x, the change in volume is proportional to what power of x?

Thanks for any suggestions on how to solve this.

I have to think more clearly about grabbing a point of a field that has a space dimension greater then one.

So that the math does not blow up, for a field of three space dimensions we grab a small spherical shell of radius R, of the field. Now we can displace this shell a distance w in the tangent space of the field. Now I think we can try to calculate volume changes.

the following function may be useful,

D(r) = A*exp[-m*r]/r for r > or = R

r is the radial distance from the center of the displaced shell, D(r) is the displacement of the field as a function of the radial coordinate r, and A is chosen so that:

D(R) = A*exp[-m*R]/R = w

Thanks for any help.

 

[Spinnor]

...
the following function may be useful,

D(r) = A*exp[-m*r]/r for r > or = R


...

Thanks for any help.

If I did the math right the radial component of the laplacian operator in spherical coordinates when operating on D(r) gives a constant squared times D(r) for r > R.

Thanks for any help!

 

[Spinnor]

And then repeat all the above questions for a massless real field,

E^2 = P^2

For a massless field use the function

D(r) = A*exp[-m*r]/r for r > or = R

and set m = 0