Re: [Phys-L] energy in the gravitational field

[John Denker <jsd@av8n.com>]

On 03/23/2015 03:00 AM, Savinainen Antti asked:

> *How* is gravitational potential energy stored in the gravitational field? 

OK, that's the hardest physics question anybody has asked in 
a long time.

Executive summary:  For most purposes, it is best to duck the
question, and just say that there is gravitational potential
energy "in the system" ... rather than trying to localize it 
some particular part of the field.

> A Finnish physics textbook answered that it is stored as mass: the
> mass of the gravitational field increases when a person lifts, say, a
> book. Do you find this explanation plausible?

That's plausible ... it's just not an explanation.  It doesn't
tell you anything you didn't already know.  In particular, it
doesn't tell you anything whatsoever about the gravitational 
field.

Consider, say, an enormous box of non-interacting particles
such as garbanzo beans ... no gravitational interaction, no
electromagnetic interaction, just free particles.  (There
must be some interaction with the walls of the box, but
that is a negligible part of the overall situation).  If
you do work on the garbanzo beans from outside the system,
the mass of the system goes up.  This is required by the 
most basic notions of special relativity, in particular 
the idea that momentum and energy form a four-vector.
The negative gorm of this 4-vector is E^2 - p^2, which 
must be some Lorentz invariant quantity, independent of 
the choice of reference frame.  We call this quantity 
the /mass/ squared.  If you increase the energy (without 
increasing the momentum), the mass goes up ... for beans 
or anything else.[**]

So for a /system/ consisting of a planet plus book, if
you do work on the system from outside, the mass of the
/system/ must go up.  That's nice and simple.

OTOH if you want to know "how" and "where" the energy is
stored, there's nothing simple about it.  For 100 years,
smart people have tried and failed to figure this out.

For the purposes of first-semester high-school physics,
one can imagine that lifting a book imparts an energy
E = m g h that resides "in" the book.  However, even 
back on Day One of modern science, Galileo was quite 
aware that such a simple approach was valid only for 
laboratory length-scales, and could not be applied to 
planetary length-scales or beyond.  By second-semester 
high-school physics we discuss Newton's law of universal 
gravitation, in which the book and the planet play 
symmetrical roles.  You could say the energy is "in" 
the planet rather than the book, and it would make 
equally much sense, i.e. no sense at all.

The next thing that everybody tries assumes there
must be energy "in the field".  If so, you ought to 
be able to write down an expression for the energy 
density in terms of the field strength ... but that 
is not easy to do.  Suppose you try to write an energy 
density proportional to g^2, in analogy to the energy 
and Poynting momentum in the electromagnetic field.
  http://www.feynmanlectures.caltech.edu/II_27.html
If you try that, you discover a nasty minus sign:  A
stronger gravitational field means /lower/ energy.
This is very unlike the electromagnetic situation, and 
is an embarrassment when you consider a gravitational
wave, which is supposed to carry positive energy.
What's worse is that the gravitational field "g" has
very limited physical significance anyway.  A uniform
g field is the same as nothing, because you can choose
a freely-falling frame in which there is no g at all.
If there is a non-uniform field, you can choose two
different frames, and one frame says the g^2 energy
is over here, while the other says it is over there.

Things just get uglier from there.

The most annoying thing is that this question SHOULD
have an answer.  For any object, you should be able
to put a test particle in orbit around it at some
radius, and infer the mass inside the orbit using
Kepler's laws.  Repeat for various radii to map out
the entire mass distribution.  That works in principle,
but it's 42 orders of magnitude too insensitive in
practice.

Bottom line: For most purposes, it is best to duck 
questions about trying to localize the gravitational 
potential energy.  Instead you can just say the
energy is "in the system" without being specific.

-------------------------

  [**] Tangential remark:  The physics described above 
  is rather imperfectly summarized by saying E=mc^2.  
  Despite being the most famous equation in the world, 
  it is very widely misunderstood, to say the least. 
  It's not even Lorentz invariant.  It is not the smart
  way to thing about the situation.  Four-vectors are
  better.

  Also note:  The mass is /not/ the only source term
  giving rise to the gravitational field.  It is the
  dominant contribution in the classical limit, but it
  cannot possibly be the only contribution in general.
  In general relativity you need to worry about the
  stress-energy tensor, which includes the energy, the
  momentum, the pressure, and shear stress.