relativistic counterpart of gravitational energy
Jim Jastrzebski
Could anyone tell me what in general relativity (GR) corresponds
to the "(potential) gravitational energy" of Newtonian gravity (NG)?
Since GR contains in itself the whole NG, which means that there
is no physics in NG that is not represented by something in GR
(however of course not vv), then what represents the Newtonian
"gravitational energy"? What is the relativistic counterpart of
Newtonian "gravitational energy"?
The tricky part is of course that while in NG we may imagine
"gravitational attractive force" and an object that gains kinetic
energy while falling "down" or losing kinetic energy while moving
"up" as exchanging the energy with something (some accumulator
of "potential energy") through "gravitational attractive force" we
can't do it in GR since we know that in free fall no forces act on
an object (neglecting tidal forces that woudn't sifice anyway) and
so we don't see immediately where the kinetic energy (that is
undoubly there) comes from or goes to. In what form that energy
is stored while it is not in a form of kinetic energy. That it does go
somewhere while the object rises and loses its kinetic energy is
obvious from the fact that we can observe cycles in which the
highest and the lowest kinetic energy is always the same (not to
mention the principle of conservation of energy which some
people may oppose and then we get into long philosophical
argument that I'd like to avoid).
So basically the question is what equation that shows relation
between measurable variables (like velocity, distance, mass, etc,
similarly like in NG) shows that the total amount of energy at
any point on e.g. eliptical stable orbit of some planet is alwas the
same. It is of course easy to demonstrate in NG introducing the
concept of "gravitational attraction" but it does not work in GR
for lack of that handy "gravitational attraction". It might be a
too simple a question for this news group but I be happy with
just a simple answer.
-- Jim
Greg Kuperberg
Jim Jastrzebski <Jim...@aol.com> wrote:
>Could anyone tell me what in general relativity (GR) corresponds
>to the "(potential) gravitational energy" of Newtonian gravity (NG)?
>Since GR contains in itself the whole NG, which means that there
>is no physics in NG that is not represented by something in GR
>(however of course not vv), then what represents the Newtonian
>"gravitational energy"? What is the relativistic counterpart of
>Newtonian "gravitational energy"?
Your question is really two questions rolled into one, because there
are two non-quantum models of gravity that come after Newtonian gravity.
The fancier, more famous, and more correct model is general relativity.
The other model is "linearized relativistic gravity" (LRG), which obeys
the rules of special relativity, but which neglects higher order effects
of curvature of spacetime. Another way to express the inaccuracy of LRG
is that "gravity begets gravity". In true gravity, the energy of the
gravity field would create extra gravitational attraction, but the
linearized theory you ignores this non-linear effect.
In linearized relativistic gravity there is a formula for the energy
density of the gravity field which is highly analogous to the formula
for the energy density of the electromagnetic field. In that theory
the answer energy density is (E^2 + B^2)/8/pi (in suitable units), where
E is the electric field and B is the magnetic field. I don't know the
particular formula for linearized gravity, but it has to be some similar
quadratic expression.
In full general relativity your question is harder and it has been the
genesis of important research in physics and geometry. The gravity field
is gone and space is curved instead, and it's not at all clear what you
might mean by its energy. However, if you have a localized (but possibly
complicated and changing) gravity well in an asymptotically flat space,
it exerts a certain amount of pull on distant objects, and therefore
it acts as if it has a certain amount of mass. This best candidate for
this mass was defined rigorously in the 1960's by Arnowitt, Deser, and
Misner, and is called ADM mass. The ADM mass includes the mass of the
things creating the gravity well (I think), as well as the effective mass
of the gravity well itself. Effective mass is equivalent to effective
potential energy just by E = mc^2. The big research question about ADM
mass is whether it is always positive. This was proved by Schoen and Yau,
and later independently by Witten.
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *
Steve Carlip
> Could anyone tell me what in general relativity (GR) corresponds
> to the "(potential) gravitational energy" of Newtonian gravity (NG)?
> Since GR contains in itself the whole NG, which means that there
> is no physics in NG that is not represented by something in GR
GR contains Newtonian gravity *as an approximation*, so you
shouldn't necessarily expect a Newtonian gravitational potential
except as an approximation. Depending on what approximation
you want to use, there can be a number of answers to this:
1. In the weak field approximation, the time-time component of
the metric (in a ``nearly rectilinear coordinate system'') depends
linearly on the Newtonian gravitational potential, and you can
read off the potential energy from that.
2. For a static gravitational field, you can always find coordinates
in which the metric looks like
ds^2 = dt^2 - (a spatial piece with time-independent coefficients)
For a test particle moving in such a field, the time component p^t
of its four-momentum is a conserved total energy, which includes
the expected kinetic energy and a ``gravitational potential energy''
piece. More invariantly, a static field has a timelike Killing vector
k_a, and (k_a)(p^a) is conserved along geodesics. Note that this
gives you total energy, not just gravitational potential energy; while
there's usually a natural-looking way to then separate out the piece
you want to call gravitational, it's not completely well-defined.
3. For the full general theory of relativity, there is no well-defined
local gravitational potential energy density. There is, however, a
``quasilocal'' energy, the energy inside an arbitrary finite region
with a specified boundary and specified motion of the boundary.
You can find this in papers by Brown and York; see, for example,
Phys. Rev. D47, 1407-1419 (1993). Here, too, what you get is a
*total* energy, from which you have to subtract off what you
consider the nongravitational part---a tricky process, since the
``gravitational'' and ``nongravitational'' pieces generally couple
to each other.
Steve Carlip
JimJast
> 1. In the weak field approximation, the time-time component
> of the metric (in a ``nearly rectilinear coordinate
> system'') depends linearly on the Newtonian gravitational
> potential, and you can read off the potential energy from
> that.
I know that. My question was not how to get numerical value of
"Newtonian potential" from GR but what real physical feature
of the real world (as described by GR) corresponds to the
"Newtonian potential" that we know is not there at all (since
there are no "gravitational attractive forces" that would
justify the existence of such potential). So it was like a
question what is the physical mechanism through which the
time-time component of the metric simulates the "Newtonian
potential".
> 2. For a static gravitational field, you can always find
> coordinates in which the metric looks like
> ds^2 = dt^2 - (a spatial piece with time-independent
> coefficients)
> For a test particle moving in such a field, the time
> component p^t of its four-momentum is a conserved total
> energy, which includes the expected kinetic energy and
> a ``gravitational potential energy'' piece.
> [snip]
> 3. For the full general theory of relativity, there is no
> well-defined local gravitational potential energy density.
> [snip]
> Here, too, what you get is a *total* energy, from which you
> have to subtract off what you consider the nongravitational
> part---a tricky process, since the ``gravitational'' and
> ``nongravitational'' pieces generally couple to each other.
Of course I know also that if the total energy is conserved I
can subtract kinetic energy from that total and get "the other
part" that I may call "relativistic counterpart of
gravitational energy" but it would be only playing with names
and not explanation of physics.
The explanation of physics would be an explanation why the
kinetic energies of free falling objects change while there
are no forces acting on those free falling objects (or in
other words an explanation why those objects have to follow
geodesics in spacetime). An explanation of a reason why the
total energy gets split into kinetic and "other" energy.
As we know the "Newtonian gravitational potential" does not
exist in the real world, but kinetic energy of the free
falling objects that seems to be real still changes, so
something else than "gravitational potential" must be
responsible for those changes. What is it? And how it keeps
energy conserved? Where the kinetic energy of the falling
object goes when it diminishes, and where it comes back from
when it increases? What is the physical nature of that
accumulator of energy that used to be that fictitious
"gravitational potential energy" of Newtonian gravity? In
short the question is about *physics* not about the
equations that describe that physics. However of course
whatever one says has to be supported by math producing the
same results the physics predicts so the explanation
wouldn't be a pure fantasy but something calculable and
verifiable. We already have math that works. I'm just asking
for explanation of physics behind this math.
-- Jim
John Baez
JimJast <jim...@aol.com> wrote:
>The explanation of physics would be an explanation why the
>kinetic energies of free falling objects change while there
>are no forces acting on those free falling objects (or in
>other words an explanation why those objects have to follow
>geodesics in spacetime). An explanation of a reason why the
>total energy gets split into kinetic and "other" energy.
>
>As we know the "Newtonian gravitational potential" does not
>exist in the real world, but kinetic energy of the free
>falling objects that seems to be real still changes, so
>something else than "gravitational potential" must be
>responsible for those changes.
No.
In situations where general relativity tells you that
"potential energy" no longer makes much sense, it
typically also tells you that "kinetic energy" and
"total energy" also don't make much sense - so the
problem you are worrying about evaporates!
This is not hard to see:
To know what "kinetic energy" is, you need to know
what "velocity" is. But velocity with respect to what?
In situations where general relativity is important,
this question typicaly has no god-given best answer.
Even in special relativity it's a bit tricky: we can
only answer it after choosing an inertial frame.
You can think of this as a field of synchronized
clocks, none accelerating relative to one another.
But when we get to general relativity there is no
such thing, typically, as an inertial frame: thanks
to gravity warping spacetime, we simply cannot create
a field of synchronized clocks.
The only case I know where kinetic energy makes sense
is the case of a static spacetime, where we can slice
spacetime up in such a way that the geometry of "space"
(the slices) does not change with the passage of "time".
In this case we can measure velocities relative to the
"unchanging fabric of space" (sorry to get a bit poetic
here, but I'm trying to avoid math), and use this to
define kinetic energy. As Steve Carlip pointed out,
this is also the case where potential energy makes sense!
And in this case, kinetic + potential energy is conserved.
I suggest that you reread Steve Carlip's reply while
keeping this fact in mind. He wasn't trying to explain
some general substitute for potential energy that works
throughout general relativity - indeed, the whole point
is that THERE IS NO SUBSTITUTE, except in special situations!
Indeed, there is no reason to believe in conservation of
energy in situations where the geometry of spacetime is
changing with the passage of time. There is even
good observational evidence against it: the cooling of
the microwave background radiation with the passage of
time. Conservation of energy is intimately linked
with symmetry under time translation, and when one goes,
so does the other. This is one of the great lessons
of modern physics.
JimJast
Future posts to this thread will be examined carefully with respect
to the charter's prohibition against "multiple posts saying the same
thing. -TB]
> In situations where general relativity tells you that
> "potential energy" no longer makes much sense, it
> typically also tells you that "kinetic energy" and
> "total energy" also don't make much sense - so the
> problem you are worrying about evaporates!
Take a case when you are standing next to a tall
building on planet earth and somone drops a brick
from the roof of that building. Can't you estimate
what kinetic energy of that brick will be when it
hits the ground, and wouldn't it be better not to
test the idea that the problem of kinetic energy
energy evaporates and remain in the path of that
brick?
Yet general relativity tells me "that 'potential
energy' no longer makes much sense" here
since in the real world (described by GR) the
earth does not attract that brick, so no force
acts on that brick during its trip down (except
some negligible air resistance and even more
negligible tidal forces). So not worrying that
the kinetic energy of that brick increases
above what it was at the roof, and assuming
that it didn't because no forces acted on the
brick might be a fatal mistake. But if this
kinetic energy (now rather well defined)
increased above what it was on the roof,
there must be some other energy that
decreased and gets somehow into that brick.
So I'm asking what this other energy is
(physically: what combination of what
variables and why) and how it got into
that brick while it was falling.
Ilja Schmelzer
> To know what "kinetic energy" is, you need to know
> what "velocity" is. But velocity with respect to what?
> In situations where general relativity is important,
> this question typicaly has no god-given best answer.
> Even in special relativity it's a bit tricky: we can
> only answer it after choosing an inertial frame.
The same "problem" you have already in Newtonian mechanics.
Ilja
--
I. Schmelzer, <il...@ilja-schmelzer.net> , http://ilja-schmelzer.net
Ralph E. Frost
news:20010327233914...@ng-ff1.aol.com...
> So I'm asking what this other energy is
> (physically: what combination of what
> variables and why) and how it got into
> that brick while it was falling.
Sum the energy expended by the mason tender who carried the brick to the
top of the building. Relate that to the amount of sunlight falling on the
wheat or corn fields from which the worker got his breakfast.
Charles Francis
<ba...@galaxy.ucr.edu> writes
>Indeed, there is no reason to believe in conservation of
>energy in situations where the geometry of spacetime is
>changing with the passage of time. There is even
>good observational evidence against it: the cooling of
>the microwave background radiation with the passage of
>time. Conservation of energy is intimately linked
>with symmetry under time translation, and when one goes,
>so does the other. This is one of the great lessons
>of modern physics.
In a related question, I have often wondered about conservation of
momentum. I don't see how it can make sense in the general case. This
seems a highly significant question because I don't know of any
theoretical argument that the universal constant of gravitation is
actually the same for all elementary particles except from conservation
of momentum. Is there one? or was Dr Stewart who taught me gtr off the
wall when he said the gravitational constant might not be truly
universal? Even quite large differences in the gravitational constant
for different elementary particles could become small when averaged over
the constituents of ordinary matter, and would be very difficult to
detect.
--
Charles Francis
John Baez
JimJast <jim...@aol.com> wrote:
>[Moderator's note: I think we're starting to go in circles here.
>Future posts to this thread will be examined carefully with respect
>to the charter's prohibition against "multiple posts saying the same
>thing. -TB]
I'll do my best not to run afoul of this wise warning.
>> In situations where general relativity tells you that
>> "potential energy" no longer makes much sense, it
>> typically also tells you that "kinetic energy" and
>> "total energy" also don't make much sense - so the
>> problem you are worrying about evaporates!
>Take a case when you are standing next to a tall
>building on planet earth and somone drops a brick
>from the roof of that building. Can't you estimate
>what kinetic energy of that brick will be when it
>hits the ground, and wouldn't it be better not to
>test the idea that the problem of kinetic energy
>energy evaporates and remain in the path of that
>brick?
Your rhetorical maneuver here consists of conflating
three completely different questions:
1) whether you'll get hurt by the falling brick
2) whether the kinetic energy of the brick is well-defined
in the limit where we neglect general-relativistic corrections
to Newtonian physics
and
3) whether the kinetic energy of the brick is well-defined
if we take all general-relativistic corrections into account.
The answer to questions 1) and 2) is yes: I don't dispute
that! But the answer to question 3) is "no".
In this particular example, the relevant corrections are
miniscule - you would have to do a very careful experiment
to see them. But they're there in principle: if people
did ultra-careful measurements of the brick's energy, they
would get slightly different answers depending on how the
experiment was done, and ultimately the only way they could
settle the question of the exact energy would be by arbitrary
fiat or (better) discarding it as ultimately meaningless.
>So I'm asking what this other energy is
>(physically: what combination of what
>variables and why) and how it got into
>that brick while it was falling.
In the Newtonian approximation to reality, this is a
perfectly fine question with an answer you already know.
If you want to understand general relativity, this is
precisely the question you must cease asking - because
it no longer makes sense when you take general-relativistic
corrections into account.
It is always surprising when it happens, but sometimes to learn
more about the world we must stop asking certain questions...
... namely, those based on false assumptions.
Charles Francis
<bpring...@YAHOO.COM> writes
>>>>>> "CF" == Charles Francis <cha...@clef.demon.co.uk> writes:
>[snip]
> CF> Even quite large differences in the gravitational constant for
> CF> different elementary particles could become small when averaged
> CF> over the constituents of ordinary matter, and would be very
> CF> difficult to detect.
>
>How do you differentiate between a `different gravitational constant'
>and a change in mass?
Although using a different gravitational constant is equivalent to a
change in gravitational mass, it is not the same as a change in inertial
mass and would result in the inequality of inertial and gravitational
mass.
> Two particles are necessary in order for a
>gravity to work. Are you saying that measuring several particles
>would not be transitive because a different constant would apply in
>each case?
No. That, in the Newtonian approximation the gravitational field
generated by different types of elementary particle is not necessarily
directly proportional to their inertial mass. Because normal matter has
a broadly similar composition of up, down quarks and electron and
binding energy (e.m. and strong) this difference would be largely
averaged out, and could be very difficult to detect.
It would mean that if you consider two gravitating bodies with different
gravitational constants in a Euclidean reference frame, then one would
attract the other more than the other would attract the one, so
conservation of momentum would not be obeyed in that reference frame. In
relativistic quantum field theory Noether's theorem conclusively
demonstrates conservation of momentum in my view, but it applies to
local interactions between particles in a flat space. I do not see how
we can extend it to apply to distant matter an artificially constructed
Euclidean reference frame in the non-Euclidean space required to treat
gravity.
>This would result in non-conservative fields (or so I
>believe).
I suspect that there may be a problem with the consistency of the
curvature field in weak interactions. But I don't know how to study it
rigorously. Hence the question. Can anyone give an answer that depends
on more than belief?
> Or is it that elementary particles do not behave the same,
>in which case are they really elementary?
We have a certain number of types of elementary particle which are
distinguished by their behaviour. If the same gravitational constant
does not apply to each then it is merely another attribute by which they
are distinguished.
--
Charles Francis
Jonathan Thornburg
Charles Francis <cha...@clef.demon.co.uk> wrote:
>in the Newtonian approximation the gravitational field
>generated by different types of elementary particle is not necessarily
>directly proportional to their inertial mass. Because normal matter has
>a broadly similar composition of up, down quarks and electron and
>binding energy (e.m. and strong) this difference would be largely
>averaged out, and could be very difficult to detect.
In other words, this would be a composition-dependent violation of
the equivalence of active gravitational mass and inertial mass. If
we accept conservation of momentum, then active gravitational mass
is necessarily the same as passive gravitational mass,
[Consider a gedanken experiment in Minkowski spacetime:
set up a dumbell whose two ends have differing active/passive
gravitational mass ratios. Say m1_p = m2_p = m_p, but
m1_a != m2_a. Then the gravitational force on m1 is
G m_p m2_a/r^2, while that on m2 is (oppositely directed)
G m_p m1_a/r^2, so the net (sum) force will be nonzero,
leading to a self-acceleration, i.e. a violation of
conservation of momentum.]
but Charles' comments were in a context where conservation of momentum
might not hold.
It's interesting to consider what experimental knowledge we have
of the possibility of such violations. That is, can we put any
experimental bounds on them? Hmm...
The Kruezer experiment shows that
active gravitational mass = passive gravitational mass
for lab-sized bodies of very different chemical composition (= relative
fractions of strong, weak, and electromagnetic binding energy, of protons,
neutrons, and electrons, of differnt quark types). I forget the accuracy
limit, but I think it was around 1e-4 or so.
A variety of earth-surface equivalence-principle and 5th-force experiments
tell us that passive gravitational mass = inertial mass
for lab-sized bodies of very different chemical composition (= relative
fractions of various stuff, as above, down to accuracy levels of 1e-12 or so.
We know from lunar laser ranging that the moon and the earth free-fall
with the same gravitational acceleration in the sun's gravitational field.
The moon and earth have significantly different chemical compositions
(same point as before)... but also significantly different relative
fractions of gravitational binding energy. Combining this with the
equivalence-principle & 5th-force experiments, we infer that
passive gravitational mass = inertial mass for gravitational
binding energy, i.e. (negative) graitational field energy.
References:
@book
{
Will,
author = "Clifford M. Will",
title = "Theory and Experiment in Gravitational Physics",
edition = "Revised",
publisher = "Cambridge University Press",
address = "Cambridge (UK)",
year = "1993",
isbn = "0-521-43973-6 (paperback)",
snote = "Only adds one new ``update'' chapter to 1st edition",
}
@unpublished
{
Will-update-1998,
author = "Clifford M. Will",
title = "The Confrontation Between General Relativity
and Experiment: A 1998 Update",
year = 1999, month = "11 November",
note = "Lecture notes from the 1998 SLAC Summer Institute
on Particle Physics,
also gr-qc/9811036
(http://arxiv.org/abs/gr-qc/9811036 and many mirrors),
also http://wugrav.wustl.edu/People/CLIFF/update98.ps ,
also audio and video of the lecture at
http://www.slac.stanford.edu/gen/meeting/ssi/1998/will.html
",
}
--
-- Jonathan Thornburg <jth...@thp.univie.ac.at>
http://www.thp.univie.ac.at/~jthorn/home.html
Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik
Q: Only 6 countries have the death penalty for children. Which are they?
A: Congo, Iran, Nigeria, (Pakistan[*]), Saudi Arabia, United States, Yemen
[*] Pakistan reportedly ended it in July 2000. -- Amnesty International
http://www.web.amnesty.org/ai.nsf/index/AMR511392000
Steve McGrew
[snip]
>In this particular example, the relevant corrections are
>miniscule - you would have to do a very careful experiment
>to see them. But they're there in principle: if people
>did ultra-careful measurements of the brick's energy, they
>would get slightly different answers depending on how the
>experiment was done, and ultimately the only way they could
>settle the question of the exact energy would be by arbitrary
>fiat or (better) discarding it as ultimately meaningless.
Bricks are pretty complicated things. It might be easier to
analyze the kinetic and gravitational potential energy of an electron.
Aside from that, however:
You've indicated that there is no meaningful way to define
either gravitational potential energy or the kinetic energy of an
object. Is there *any* conserved quantity in general relativity that
corresponds to total energy of a system? If so, what is it, and can
it be calculated from the motion of the particles and their relative
positions? I can imagine that if the two terms (motion and position)
are not separable, then it would be possible to have a conserved total
energy without being able to point at one term and meaningfully call
it "kinetic energy" and another term and call it "gravitational
potential energy".
Steve
Charles Francis
<jth...@galileo.thp.univie.ac.at> writes
>
>The Kruezer experiment shows that
>active gravitational mass = passive gravitational mass
>for lab-sized bodies of very different chemical composition (= relative
>fractions of strong, weak, and electromagnetic binding energy, of protons,
>neutrons, and electrons, of differnt quark types). I forget the accuracy
>limit, but I think it was around 1e-4 or so.
Can you describe the experiment?
>
>A variety of earth-surface equivalence-principle and 5th-force experiments
>tell us that passive gravitational mass = inertial mass
>for lab-sized bodies of very different chemical composition (= relative
>fractions of various stuff, as above, down to accuracy levels of 1e-12 or so.
I am fully convinced of the equivalence of passive gravitational mass
and inertial mass. It is active gravitational mass that I think should
be investigated as deeply as possible. 1e-4 is quite good, but I think
not conclusive because of the averaging effect. I did a calculation
based on a ration of 3:2:1 for the gravitational constants of electron,
up down, with no allowance made for gravitional effect of the strong and
e.m. fields in the atom and got 1e-2 as a maximum expected difference to
be found in different elements. That could be entirely lost in the
gravitational effect of the strong field, if as we think the bare masses
of the u and d are of the order of the electron mass and most of the
mass of the proton and neutron is in the binding energy.
--
Charles Francis
John Baez
Steve McGrew <ste...@iea.com> wrote:
>On 1 Apr 2001 04:55:58 GMT, ba...@galaxy.ucr.edu (John Baez) wrote:
>>In this particular example, the relevant corrections are
>>miniscule - you would have to do a very careful experiment
>>to see them. But they're there in principle: if people
>>did ultra-careful measurements of the brick's energy, they
>>would get slightly different answers depending on how the
>>experiment was done, and ultimately the only way they could
>>settle the question of the exact energy would be by arbitrary
>>fiat or (better) discarding it as ultimately meaningless.
> Bricks are pretty complicated things. It might be easier to
>analyze the kinetic and gravitational potential energy of an electron.
You can substitute "electron" for "brick" throughout the above
paragraph if you like; it's still true.
> You've indicated that there is no meaningful way to define
>either gravitational potential energy or the kinetic energy of an
>object.
Except in certain special cases, or in certain approximations.
>Is there *any* conserved quantity in general relativity that
>corresponds to total energy of a system?
No, and this actually what I was trying to say in the passage you
quoted. In general relativity there is no conserved quantity
deserving the name "energy"... except in certain special cases, or
in certain approximations.
If you're wondering why I keep sticking in those caveats, it's
because there are lots of nuances here to worry about...
... AFTER one has grasped the shocking basic point that conservation
of energy - in the naive form we all know and love - gets THROWN OUT
THE WINDOW in general relativity!
It gets thrown out for the very simple reason that conservation of
energy is a consequence of time translation symmetry, and this goes
away when the geometry of space changes drastically with time, as it
does in the big bang cosmology, for example.
In situations where you can get some sort of (possibly approximate
or watered-down version of) time translation symmetry, you get some
(possibly approximate or watered-down version of) conservation of
energy. For a little run-down of some of these situations, see:
http://math.ucr.edu/home/baez/physics/energy_gr.html
Charles Francis
me
In article <9aidea$k31$1...@woodrow.ucdavis.edu>, Steve Carlip
<car...@dirac.ucdavis.edu> writes
>
>Which makes 5x10^{-5} a very strong limit, right? If you look at the
>details, nuclear electrostatic binding energy accounts for about 4.8%
>of the mass of bromine, and only about .8% of the mass of fluorine. If
>this energy didn't gravitate, the results would disagree with the Kreuzer
>limit by a factor of around 1000.
I think that is conclusive. Chargeless virtual photons definitely have
active gravitational mass. Thanks. I do like to be sure that fundamental
aspects of physical theory are either proven or marked for
investigation.
--
Charles Francis
Jonathan Thornburg
| The Kruezer experiment shows that
| active gravitational mass = passive gravitational mass
| for lab-sized bodies of very different chemical composition (= relative
| fractions of strong, weak, and electromagnetic binding energy, of protons,
| neutrons, and electrons, of differnt quark types). I forget the accuracy
| limit, but I think it was around 1e-4 or so.
In article <9ailhd$7lp$1...@news.state.mn.us>,
Charles Francis <cha...@clef.demon.co.uk> asked:
>Can you describe the experiment?
I am currently 6 time zones away from my GR books and articles, so
this will be from memory. And it's been 15+ years since I looked at
the original Kreuzer paper. Having said that, here goes...
Kreuzer (I think I have the spelling right, but I'm not 100% sure)
set up a horizontal tube containing a teflon mass floating in some
organic liquid, I think tribromoethelyene or somthing similar. The
liquid was heated so the teflon mass was at neutral bouyancy.
Then he used this whole assembly as the "big mass" in a Cavendish
balance, and moved the teflon mass back and forth in the tube.
Because the teflon mass was floating at neutral bouyancy, he knew it
had the same passive gravitational mass (--> weight in the Earth's
gravitational field) as the volume of fluid it displaced. So if the
teflon mass's ratio of active/passive gravitational mass differed from
that of the fluid, the net force on the Cavendish balance would change
when he teflon mass was moved. Kruezer measured the motion of the
Cavendish balance, and didn't see any sign of such an effect to within
the error limits of his data.
The Kreuzer experiment thus sets an upper limit on the diffrence in
active/passive gravitational mass for teflon and tribomoethylene.
These materials have very different chemical compositions (eg they
contain lots of chlorine and bromine respectively). One very interesting
conclusion we can draw from this was stated by Steve Carlip
<car...@dirac.ucdavis.edu> in article <9aidea$k31$1...@woodrow.ucdavis.edu>
(in another thread):
# nuclear electrostatic binding energy accounts for about 4.8%
# of the mass of bromine, and only about .8% of the mass of fluorine. If
# this energy didn't gravitate, the results would disagree with the Kreuzer
# limit by a factor of around 1000.
For further details on the Kreuzer experiment and what we can learn from
it, or more generally the whole question of "the equivalence principle"
and experimental tests of it, the canonical reference is
@book
{
Will,
author = "Clifford M. Will",
title = "Theory and Experiment in Gravitational Physics",
edition = "Revised",
publisher = "Cambridge University Press",
address = "Cambridge (UK)",
year = "1993",
isbn = "0-521-43973-6 (paperback)",
snote = "Only adds one new ``update'' chapter to 1st edition",
}
This includes references to many original papers, including Kreuzer's.
--
-- Jonathan Thornburg <jth...@thp.univie.ac.at>
http://www.thp.univie.ac.at/~jthorn/home.html
Universitaet Wien (Vienna, Austria) / Institut fuer Theoretische Physik
"A puritan is someone who is really afraid that
someone somewhere is having fun." -- Norbert Roth
Steve McGrew
wrote:
>In article <3acaaad8...@nntp.iea.com>,
>Steve McGrew <ste...@iea.com> wrote:
>>Is there *any* conserved quantity in general relativity that
>>corresponds to total energy of a system?
[snip]
>... AFTER one has grasped the shocking basic point that conservation
>of energy - in the naive form we all know and love - gets THROWN OUT
>THE WINDOW in general relativity!
>
>It gets thrown out for the very simple reason that conservation of
>energy is a consequence of time translation symmetry, and this goes
>away when the geometry of space changes drastically with time, as it
>does in the big bang cosmology, for example.
>
>In situations where you can get some sort of (possibly approximate
>or watered-down version of) time translation symmetry, you get some
>(possibly approximate or watered-down version of) conservation of
>energy. For a little run-down of some of these situations, see:
>
>http://math.ucr.edu/home/baez/physics/energy_gr.html
Your article at http://math.ucr.edu/home/baez/physics/energy_gr.html
indicates that the complication resides in the challenge of how to
compare two energy-momentum vectors at different locations in
spacetime: if one could put one vector in a box and carry it to the
other location for side-by-side comparison, the results of comparison
would depend on the path followed from the first location to the
second.
I'm curious to know if a sum-over-all-paths approach has been tried
for comparing two vectors at different locations. For a brick it would
be pretty difficult, but perhaps for an electron or a photon it might
be possible.
Steve
John Baez
Steve McGrew <ste...@iea.com> wrote:
I won't attempt to answer this in detail; I'll just say two things
about it.
1) If you drop a brick, its behavior is in fact described by
a sum over all paths, with each path weighted by the complex
amplitude exp(iS), where S is the path's action.
This is what quantum mechanics says. The fact that classically
the brick appears to trace out a geodesic is a consequence of the
fact that there's tremendous cancellation in this sum except near
the paths that are stationary points of the action: the geodesics.
If you drop a spinning brick, classically its angular momentum vector
gets "Fermi-Walker transported" along the geodesic it traces out.
Fermi-Walker transport is a cousin of parallel transport.
Quantum mechanically, we see that this Fermi-Walker transport is
really just an approximation to a sum over all possible ways the
spinning brick could move.
2) Diffusion of heat is also described by a sum over all paths -
"Brownian motion of heat particles", so to speak. I know there's
no such thing as heat particles, but it's a handy metaphor!
Heat particles are scalars - i.e., the temperature field is a
scalar field. If instead we used vectors, we'd get "vector heat",
which would diffuse via the heat equation for vector fields. You
can think of as a way of transporting vectors via a sum over all paths.
JimJast
>of energy - in the naive form we all know and love - gets THROWN OUT
>THE WINDOW in general relativity!
>
>It gets thrown out for the very simple reason that conservation of
>energy is a consequence of time translation symmetry, and this goes
>away when the geometry of space changes drastically with time, as it
>does in the big bang cosmology, for example.
>
Wouldn't it be more reasonable to keep GR and the
conservation of energy but throw out the window
the big bang cosmology and adopt one with time
translation symmetry instead?
[Moderator's note: The key phrase here is "for example": the
difficulty reconciling energy conservation with general relativity is
not restricted to big-bang models. If you want both energy
conservation and GR, you have to throw away all situations without
time translation symmetry -- in other words, all situations in which
anything ever happens! -TB]