Real relativistic field and curvature?

In summary, the conversation discusses a real field that satisfies E^2 = P^2 + m^2, with the assumption of 4D spacetime and a stationary field. The question is posed if a 3D "anchored" membrane will change its volume if a single point is displaced, with a proposed function to calculate volume changes. The conversation also mentions repeating the questions for a massless real field with the suggested function D(r) = A*exp[-m*r]/r for r > or = R and setting m = 0.
  • #1
Spinnor
Gold Member
2,226
431
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.
 
Last edited:
Physics news on Phys.org
  • #2
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.
 
  • #3
And then we would like to let the point go and picture volume changing with time and space. But first things first.
 
  • #4
And then repeat all the above questions for a massless real field,

E^2 = P^2
 
  • #5
Spinnor said:
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.
 
  • #6
Spinnor said:
...
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!
 
  • #7
Spinnor said:
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
 

FAQ: Real relativistic field and curvature?

1. What is a real relativistic field?

A real relativistic field is a concept in physics that describes how energy and matter interact and behave in the presence of gravity. It is a fundamental part of Einstein's theory of general relativity and is used to explain the curvature of spacetime.

2. How does real relativistic field differ from classical field theory?

Classical field theory, also known as Newtonian field theory, describes the behavior of objects in a flat, non-curving space. Real relativistic field, on the other hand, takes into account the effects of gravity and describes the behavior of objects in a curved space.

3. What is curvature in the context of real relativistic field?

In the context of real relativistic field, curvature refers to the bending of spacetime caused by the presence of massive objects. This curvature is what we experience as gravity and is described mathematically by Einstein's field equations.

4. How is real relativistic field used in practical applications?

Real relativistic field has many practical applications, including in the fields of astronomy, cosmology, and space exploration. It is used to accurately predict the behavior of objects in the presence of strong gravitational fields, such as those around black holes. It is also used in technologies such as GPS, which relies on Einstein's theory of relativity for accurate timekeeping.

5. What are some current research topics related to real relativistic field and curvature?

Some current research topics in this field include studying the effects of gravitational waves on the curvature of spacetime, investigating the role of dark energy in the expansion of the universe, and exploring the possibility of using real relativistic field to create new forms of energy. Other areas of research include testing the predictions of general relativity in extreme environments, such as near the event horizon of a black hole, and using real relativistic field to improve our understanding of the early universe.

Back
Top