General Question about Gravitational Potential & General Relativity

In summary, there is a potential quantity in general relativity analogous to Newton's theory of gravitation.
  • #1
Vectronix
64
2
Is there a potential quantity in general relativity, analogous to Newton's theory of gravitation? I am not too familiar with GR, so I thought I'd ask.
 
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  • #2
In GR, the gravitational field is the metric field, which is a tensor field.

In a class of spacetimes called "static", the metric can be written in terms of a scalar field which is analogous to the Newtonian potential.
 
  • #3
Well there's an effective potential which can be defined in GR for calculating orbits around black holes and the like.
 
  • #4
atyy said:
In GR, the gravitational field is the metric field, which is a tensor field.

In a class of spacetimes called "static", the metric can be written in terms of a scalar field which is analogous to the Newtonian potential.
Shouldn't curvature also not be weak to do this? So,

[tex]
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}
[/tex]

where the components of h are much smaller than 1?
 
  • #5
You can define the "force of gravity" in a static space-time as the proper acceleration of a stationary particle. Though you might have to define a preferred coordinate system to define the notion of a static particle. The issue I'd be concerned with is eliminating circular orbits as not being a "stationary particle".

You can also write this force as the gradient of a scalar potential. See Wald pg 158 problem 4. However, it won't follow Gauss's law, the surface integral of the "force" around an enclosing body won't be constant. However, it turns out (see Wald around pg 288)that if you do a Gauss-law like intergal on the "force at infinity" rather than the force, you do get a constant quantity that's proportional to the enclosed Komar mass.

You can also look at http://en.wikipedia.org/wiki/Komar_mass though Wald is a better source and the inspiration for this calculation.

The force at infinity is just the force mulltiplied by the redshift factor. The redshift factor can be defined as [itex]\sqrt{\xi^a \xi_a}[/itex] where [itex]\xi^a[/itex] is the timelike Killing vector of the static system. Using sensible coordinates, which you probably need to define the notion of a stationary particle anyway, the timelike Killing vector will just be a unit vector [1,0,0,0] and the redshift factor is equal to [itex]\sqrt g_{tt}[/itex], i.e. the square root of the time dilation factor.

For the Schwarzschild metric, the effective potential (in geometric units) whose gradient yields the force is ln(g_tt), the effective potential whose gradient yields the force-at-infinity is sqrt(g_tt).

As handy as all this is for static space-times, you can't generalize it - in general, there's no such thing as gravitational potential in GR.
 
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1. What is gravitational potential?

Gravitational potential is a measure of the amount of energy that an object has due to its position in a gravitational field. It is the potential energy per unit mass of an object at a certain point in a gravitational field, and is measured in units of energy per unit mass (such as joules per kilogram).

2. How is gravitational potential related to general relativity?

In general relativity, gravitational potential is one of the key components of the theory. It is used to describe how matter and energy affect the curvature of spacetime, which in turn determines the motion of objects in a gravitational field. This allows general relativity to explain the effects of gravity on a larger scale, such as the motion of planets and galaxies.

3. What is the difference between gravitational potential and gravitational potential energy?

Gravitational potential and gravitational potential energy are closely related concepts, but they are not the same thing. Gravitational potential is a measure of the energy per unit mass at a certain point in a gravitational field, whereas gravitational potential energy is the total energy that an object has due to its position in a gravitational field. In other words, gravitational potential is a property of the field itself, while gravitational potential energy is a property of the object within the field.

4. How is gravitational potential calculated?

Gravitational potential can be calculated using the equation V = -GM/r, where V is the gravitational potential, G is the universal gravitational constant, M is the mass of the object producing the gravitational field, and r is the distance from the object to the point where the potential is being measured. This equation assumes a spherically symmetric mass distribution, and can be used to calculate the potential at any point in the field.

5. Can gravitational potential be negative?

Yes, gravitational potential can be negative. This is because gravitational potential is a relative quantity, meaning it is measured relative to a certain reference point. If the reference point is chosen to be at infinity, then the potential at any point closer to the object will be negative. This negative potential represents a decrease in the potential energy of an object as it moves closer to the source of the gravitational field.

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