# I How is potential energy described in GR?

#### Karl Coryat

Summary
Asking how GR treats gravitational potential energy, which is typically described in a Newtonian context.
Hi, folks. Several years ago I made a YouTube video with a new demonstration of GR for a general audience ("How Gravity Makes Things Fall"). It won a pretty prestigious physics award. I still get comments and questions on it. One today stumped me: "How do we account for potential energy if gravity is not a force field?"

I consulted my Hartle textbook, but the index said, "See Newtonian gravity." There, Hartle talks about how gravitational potential is an analog of electrostatic potential. So that won't help me answer the viewer's question.

How is gravitational potential energy described within the context of curved spacetime? In Newtonian gravity, we describe a test particle's potential as a specific quantity, in terms of masses and distance. But in GR, it seems that a test particle in a curved spacetime could be described as having various potentials, depending upon the masses and distance associated with the curvature. So is PE meaningless in the context of GR?

Thank you very much for your help!

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#### Orodruin

Staff Emeritus
Homework Helper
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Generally, you don’t use the concept. In the stationary weak field limit, you can identify the Newtonian gravitational potential with the small perturbations to the flat Minkowski metric and there are some other situations where it might be applicable as a similar concept, but for the most part it is not.

#### Karl Coryat

An object's kinetic energy factors into the stress–energy tensor, so by the conservation of mass–energy, it seems that kinetic energy must "go somewhere" in GR terms, when for example an outbound comet slows down. Does the resulting potential energy contribute to the local energy density? Or is the comet's kinetic energy borrowed from the global energy density of the gravitational field?

#### pervect

Staff Emeritus
An object's kinetic energy factors into the stress–energy tensor, so by the conservation of mass–energy, it seems that kinetic energy must "go somewhere" in GR terms, when for example an outbound comet slows down. Does the resulting potential energy contribute to the local energy density? Or is the comet's kinetic energy borrowed from the global energy density of the gravitational field?
You are correct in your example of the comet that in order for energy to be conserved , one needs to account somehow for the change in kinetic energy of the comet as it slows down. The difficulty arises in the seemingly trivial idea that the energy, that does need to be accounted for somehow, must have a location.

Unfortunately, it is not possible to find a formulation for "where" exactly the energy goes. In the language of tensors, there is no tensor that describes the gravitational field energy.

There is some discussion of this in MTW's "Gravitation". See in particular section \$20.4, "Why the energy of the gravitational field cannot be localized". There is more discussion of the concepts of energy in GR Wald, "General Relativity". It's a rather advanced topic, the modern interpretation is to look at asymptotic flatness of the space-time as one of the key feature needed to define a conserved system energy.

There are a bunch of attempts to address the issue of energy in GR in the literature, more than I can get into. You might start with ADM mass and Bondi mass, if you're curious. Wald formulates both of these concepts in terms of the asymptotic properties of the space-time "at infinity", which I would point to as the modern approach. Earlier literature contains a bunch of other approaches, for instance the ADM mass originated from a Hamiltonian formalism. You'll see other approaches if you dig into it as well, such as pseudotensors. None of the approaches, as far as I am aware, involve any generalizable concepts of "potential energy", which seems to be a Newtonian idea that isn't useful in GR.

#### cosmik debris

Does the metric tensor of GR in some way take the place of potential?

Cheers

#### PeterDonis

Mentor
An object's kinetic energy factors into the stress–energy tensor, so by the conservation of mass–energy, it seems that kinetic energy must "go somewhere" in GR terms, when for example an outbound comet slows down.
It's important to distinguish two different ways of modeling this kind of scenario.

In the model usually used in practice, the comet's stress-energy is neglected (which works well in practice, since its stress-energy is in fact negligible when we are only concerned with its orbit), and it is treated as a test object orbiting in the spacetime geometry of the solar system. In the simplest such model, where we ignore all the planets and just consider the Sun, the geometry is stationary and we can define a potential energy, and we can use the usual Newtonian-type analysis of the comet's energy shifting back and forth between kinetic and potential as it orbits (though some of the details in GR will differ from the Newtonian case). The only stress-energy present in this case is in the central mass (e.g., the Sun), and it is at rest, so nothing about its stress-energy is changing.

In more complicated situations where we can't neglect the stress-energy of the orbiting object (for example, if we are looking at one of the pulsars in a binary pulsar system), the spacetime geometry is not stationary and we can't define a potential energy. In fact we generally can't find a closed form solution for the system at all and have to model it using numerical simulations. But heuristically, the stress-energy of the object changes with time, and the spacetime geometry around it changes in concert in such a way that the covariant divergence of the stress-energy tensor is zero.

Does the metric tensor of GR in some way take the place of potential?
Since the metric tensor affects the covariant divergence of the stress-energy tensor, thinking of it as taking the place of the potential is ok as far as it goes; but unfortunately it won't get you very far with the details.

#### pervect

Staff Emeritus
Does the metric tensor of GR in some way take the place of potential?

Cheers
In linearized theory, I've seen the metric coefficients referred to as a potentials. This was in the context of comparing them to the Lienard-Wiechert potentials, but I don't recall the details any more, unfortunately.

#### m4r35n357

Summary: Asking how GR treats gravitational potential energy, which is typically described in a Newtonian context.
I think Schwartzschild is covered in MTW "the pit in the potential" (ch.25) and in Carroll's GR notes, isn't it? Both of them compare and contrast with the Newtonian case.

[EDIT] here is an animation, and here is my take (GitHub).

For the Kerr spacetime, Wilkins does a similar thing.

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#### PeterDonis

Mentor
"the pit in the potential"
The potential here is not quite the same as potential energy; it's an effective potential for objects in free-fall orbits based on using constants of geodesic motion to simplify the equations. But potential energy in stationary spacetimes is a more general concept that also applies to objects that are not in geodesic motion.

#### Mentz114

Gold Member
Summary: Asking how GR treats gravitational potential energy, which is typically described in a Newtonian context.

Hi, folks. Several years ago I made a YouTube video with a new demonstration of GR for a general audience ("How Gravity Makes Things Fall"). It won a pretty prestigious physics award. I still get comments and questions on it. One today stumped me: "How do we account for potential energy if gravity is not a force field?"

I consulted my Hartle textbook, but the index said, "See Newtonian gravity." There, Hartle talks about how gravitational potential is an analog of electrostatic potential. So that won't help me answer the viewer's question.

How is gravitational potential energy described within the context of curved spacetime? In Newtonian gravity, we describe a test particle's potential as a specific quantity, in terms of masses and distance. But in GR, it seems that a test particle in a curved spacetime could be described as having various potentials, depending upon the masses and distance associated with the curvature. So is PE meaningless in the context of GR?

Thank you very much for your help!
In GTR, motion is described kinematically as opposed to dynamically. A loose analogy is to imagine that the gravitational field is four 'magnetic' fields, one for each dimension. The components of a 4-velocity interact with the fields like a charged particle does to a magnetic field. The change in velocity depends on $dx_\mu/d\tau$. This also explains why an object on a geodesic keeps moving because $dt/d\tau$ is never zero. Also $dt/d\tau$ is likened to a potential and sometimes plays that role.

The final link is that the 'magnetic' fields are described by the Ricci rotation coefficients which determine what rotations (including Lorentz boosts) will be caused by what velocities.

A very good reference is ( more than 900 pages !)

General Relativity, Black Holes, and Cosmology
Andrew J. S. Hamilton

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"How is potential energy described in GR?"

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