# So what has happened to the GPE (Gravitational Potential Energy)?

#### Xeinstein

Classically if you lift up a body, say a brick, you are doing work against the gravitational force, this work is then 'stored' as Gravitational Potential Energy and released when the body is dropped.

Energy is conserved so GPE (and other forms of PE) + KE = Constant.

In GR the problem is that there is no gravitational force as such. Gravitation is explained as the effect of space-time curvature. Free-falling bodies travel along straight lines (geodesics) through curved space-time. Like two ants crawling across the dip around the stalk of an apple, the geodesics of a dropped brick and the Earth converge because the Earth's mass (and to a much lesser extent the brick's) has curved the space-time through which they both 'fall'.

The force of 'weight' is a non-inertial force perturbing the brick from its free-falling geodesic. Release it and the brick suffers no weight at all, it is weightless.

The problem arises when you try to work out where the energy used to lift the brick goes to.

Lift a brick and put it on a shelf. The rest energy of the brick has not changed, so where has the GPE, the work expended lifting it, gone to?

The standard answer is "into the field", the presence of the brick higher up in the Earth's gravitational field has altered it slightly, the components of the GR Riemannian tensor describing that field has changed.

However if the brick now falls off the shelf it is momentarily still stationary but accelerating downwards. In that free-falling state the space-time locally around becomes flat. This is called the Einstein Equivalence Principle, "the physics of a small enough volume around a free-falling test particle becomes indistinguishable from that of SR", its coordinate system locally is Lorentzian.

So what has happened to the GPE now?

Related Special and General Relativity News on Phys.org

#### yuiop

This paper http://arxiv.org/PS_cache/hep-ph/pdf/0010/0010120v2.pdf posted by pmb_phy in a similar thread states

"The rest energy E0 of any massive object increases with increase of the distance from the gravitating body because of the increase of the potential "

So (by the logic of the paper) if a brick is thrown upwards with velocity v, then its initial total energy (TE) is mc^2*gamma(v) , its initial kinetic energy (KE) is mc^2*gamma(v)-mc^2 and its initial rest energy (RE) is TE-KE = mc^2 and . If the total energy is conserved, then when the brick reaches its peak height, its velocity and therefore its KE is momentarily zero. The rest energy is then TE-KE = TE = mc^2 gamma(v).

It should be stressed that the paper is referring to coordinate measurements made by an observer "at infinity".

We can come to the same conclusion by using the relationship $$RE = \sqrt{TE^2 - (pc)^2}$$ and assuming that TE is conserved. However, RE is supposed to be an invariant quantity in relativity, so in that case TE is not conserved and is simply lost as GR does not have the luxury of storing energy in a gravitational field. hmmmmm

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#### pervect

Staff Emeritus
Unfortunately, some of Pete's views here are not quite mainstream.

The most accurate (and perhaps surprising) answer is that it is not possible to define the "location" of the total energy of a system in general in GR. In fact, it is not even possible in the most general cases to even define what the total energy of a system is!

In important special cases, it is possible to define the total energy of a system in GR. There are actually several sorts of ways to define total energy, not just one. The ADM energy of a system is one of the more powerful ways of defining the totla energy, and has the feature that while the total energy can be defined, the location of the energy cannot be defined in a covariant manner.

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

talks about the impossibility of localizing energy in GR, as do other sources.
Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein [3], he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

talks in general about what energy conseration in General Relativity actually means - it is not as straightforwards as it is in Newtonian physics, it is probably not really possible to understand the GR viewpoint unless one actually understands GR, people with a Newtonian viewpoint shoujld simply understand that the problem is a complex one.

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#### Xeinstein

Unfortunately, some of Pete's views here are not quite mainstream.

The most accurate (and perhaps surprising) answer is that it is not possible to define the "location" of the total energy of a system in general in GR. In fact, it is not even possible in the most general cases to even define what the total energy of a system is!

In important special cases, it is possible to define the total energy of a system in GR. There are actually several sorts of ways to define total energy, not just one. The ADM energy of a system is one of the more powerful ways of defining the totla energy, and has the feature that while the total energy can be defined, the location of the energy cannot be defined in a covariant manner.

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

talks about the impossibility of localizing energy in GR, as do other sources.

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

talks in general about what energy conseration in General Relativity actually means - it is not as straightforwards as it is in Newtonian physics, it is probably not really possible to understand the GR viewpoint unless one actually understands GR, people with a Newtonian viewpoint shoujld simply understand that the problem is a complex one.
I think that's a bit strange. If total energy of a system is Not defined, then how could gravitational wave carry energy away from its source?
When a binary pulsar emits gravitational radiation, it loses orbital energy and angular momentum, which causes the orbit to shrink and the period to decrease. This decrease is what is predicted by general relativity and indicate that Gravitational Potential Energy is reduced, so we know that gravitational radiation is being emitted, since the total energy is conserved, the radiated energy must come from the GPE of the binary pulsar

Interestingly, In his book, Gravity from the Ground Up, Schutz indicate that it's possible to define the conserved energy of a particle. If the geometry is time-independent, then the total energy of a material system, as measured by an experimenter very far away, is conserved, independent of time.

Page 234, "Gravity from the Ground Up"

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#### pervect

Staff Emeritus
I think that's a bit strange. If total energy of a system is Not defined, then how could gravitational wave carry energy away from its source?
When a binary pulsar emits gravitational radiation, it loses orbital energy and angular momentum, which causes the orbit to shrink and the period to decrease. This decrease is what is predicted by general relativity and indicate that Gravitational Potential Energy is reduced, so we know that gravitational radiation is being emitted, since the total energy is conserved, the radiated energy must come from the GPE of the binary pulsar
Energy radiated away from a system can be understood in a couple of different ways. The most common approach is to consider the Bondi mass of the system - which requires an asymptotically flat space-time in order to be defined.

Another approach, which applies only to weak fields, is the approach of the effective stress-energy tensor associated with gravitational radiation. Here is a quote from MTW pg 955 on the topic.

Exercise 18.5 showed tht in principle one can build detectors which extract energy from gravitaional waves. Hence it is clear that the waves must carry energy.

Unfortunately, to derive and justify an expression for their energy requires a somewhat more sophisticated viewpoint than linearized theory. Such a viewpoint will be developed later in this chapter. (note: it's called the shortwave formalism and is on pg 964)....

The stress-energy carried by gravitational waves cannot be localized inside a wavelength. One cannot say whether the energy is carried by the crest of a wave, by its trough, or by its "walls." However, one _can_ say that a certain amount of stress-energy is contained in a given "macroscopic" region (region of several wavelength's size) and one can thus talk (note: talk about in the weak field limit!) about a tensor for an _effective_ smeared-out stress-energy of gravitational waves.

Interestingly, In his book, Gravity from the Ground Up, Schutz indicate that it's possible to define the conserved energy of a particle. If the geometry is time-independent, then the total energy of a material system, as measured by an experimenter very far away, is conserved, independent of time.

Page 234, "Gravity from the Ground Up"
This is in agreement with the sci.physics.faq

In certain special cases, energy conservation works out with fewer caveats. The two main examples are static spacetimes and asymptotically flat spacetimes.
However, it is possible to imagine space-times that are neither static nor asymptotically flat. The FRW space-time representing our universe is an important example of a space-time which has neither property.

The energy or mass directly associated with a static spacetime is called the Komar mass. Static space-times posess a time translation symmetry. Noether's theorem implies that this symmetry gives rise to a conserved energy.

It is important to know that there are useful defintions of energy in GR. It's equally important to know that there is more than one such defintion, and that none of these defintions are completely general, all of them apply only to special cases.

The big three defintions for total enery in GR are Bondi energy, ADM energy, and Komarr energy, all of which are different. To make life interesting, here are also various expressions for quasi-local defintions of energy as well. But quasi-local defintions don't give a defiinte answer for the total energy of a system. The situation is inherently more complicated in GR than it is in Newtonian physics. Noether's theorem is probably the best and simplest route to appreciate why there is such a difference.

#### yuiop

The energy or mass directly associated with a static spacetime is called the Komar mass. Static space-times posess a time translation symmetry. Noether's theorem implies that this symmetry gives rise to a conserved energy.

It is important to know that there are useful defintions of energy in GR. It's equally important to know that there is more than one such defintion, and that none of these defintions are completely general, all of them apply only to special cases.

The big three defintions for total enery in GR are Bondi energy, ADM energy, and Komarr energy, all of which are different. To make life interesting, here are also various expressions for quasi-local defintions of energy as well. But quasi-local defintions don't give a defiinte answer for the total energy of a system. The situation is inherently more complicated in GR than it is in Newtonian physics. Noether's theorem is probably the best and simplest route to appreciate why there is such a difference.
An ideal non rotating massive body (Shwarzchild geometry) satisfies the conditions of being static and asymptotically flat.

If a small brick with rest mass m_o, is thrown upwards with velocity v then its initial total energy is m_o/sqrt(1-v^2/c^2), its initial kinetic energy is m_o/sqrt(1-v^2/c^2)-m_o and its momentum is m_o v/sqrt(1-v^2/c^2)

At the top of its trajectory before it starts falling downwards the velocity and momentum of the brick is momentarily zero and so is its kinetic energy.

At the top of the trajectory what is:

The rest mass of the brick?

The rest energy of the brick?

The Komar mass of the brick?

#### Denton

I was under the impression that einstein thought that since the space between particles has either expanded or contracted, energy would be rightly converted from this process. So the energy is stored in the space itself?

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