Gravitational force and Lagrangian Mechanics
Pmb
matter of opinion only. Its is entirely based on what one calls "force."
For a particle only acted upon by a gravitational force, the 4-force
vanishes. The 3-force does not.
However there are other ways to GR than 4-vectors. There is Lagrangian
mechanics. In Lagrangian mechanics there is the generalized (aka canonical)
4-force F_u defined either as (let & = partial derivative sign)
F_u = &L/&x^u
This can equivalently be defined through generalized 4-momentum P_u defined
as
P_u = &L/&U^u
Where U^u = gamma(c,v) = gamma(c, dx, dy, dz) are the components of
4-velocity. From Lagrange's equations (let T = proper time)
d/dT( &L/&U^u) = &L/&x^u
we obtain
d/dT(P_u) = &L/&x^u = F_u
I.e.
F_u = dP_u/dT
For a particle in free-fall F_u is sometimes called the "Gravitational
force." However since
dP/dT = (dt/dT) dP_u/dt
some people call dP_u/dt the gravitational force. I (and Mould - "Basic
Relativity) prefer the later. Ohanian & Ruffini ("Gravitation and
Spacetime") prefer the later. The reason for this choice is rooted in Mach's
Principle. In Newtonian mechanics inertial forces (e.g. the Coriolis force)
is sometimes (not always) though as "fictitious." Einstein considered
inertial forces to be "real" since he considered the gravitational force to
be real i.e. caused be a source. The Coriolis force has its source in the
gravitational interaction of the "distance stars".
Therefore in general relativity the gravitational force is the *generalized
force* from Lagrangian mechanics. This is the force that Will speaks of in
his book "Was Einstein Right?"
This is also the force that Einstein speaks of in his text "The Meaning of
Relativity." For example: On page 100 if this book Einstein discusses
Mach's Principle
*******************************
...., the theory of relativity makes it appear probable that Mach was on the
right road in his thought that inertia depends on a mutual action of matter.
For we shall now show in the following that, according to our equations,
inert masses do act upon each other in the sense of the relativity of
inertia, even if only feebly. What is to be expected along the line of
Mach's thought?
1. The inertia of a body must increase when ponderable masses are piled up
in its neighbourhood.
2. A body must experience an accelerating force when neighboring masses are
accelerated, and, in fact, the force must be in the same direction as the
acceleration.
3. A rotating hollow body must generate inside of itself a "Coriolis
field," which deflects moving bodies in the sense of rotation, and a radial
centrifugal field as well.
We shall now show that these three effects, which are to be expected in
accordance with Mach's ideas, are actually present according to our theory,
although their magnitude is so small that confirmation of them in laboratory
experiments is not to be thought of.
*******************************
I agree entirely with Einstein for the reasons he outlines. The
"accelerating force" is the gravitational force dP_u/dt I explained above.
It is for the above reasons that some relativists hold that there is a
gravitational force and that it is a real force.
All comments welcome.
Pmb
Ken S. Tucker
>It is often said that gravity is not a force in GR. That, of course, is a
>matter of opinion only. Its is entirely based on what one calls "force."
>
>For a particle only acted upon by a gravitational force, the 4-force
>vanishes. The 3-force does not.
>
>However there are other ways to GR than 4-vectors. There is Lagrangian
>mechanics. In Lagrangian mechanics there is the generalized (aka canonical)
>4-force F_u defined either as (let & = partial derivative sign)
>
>F_u = &L/&x^u
That's a good example, where F_u =0 in GR
when L = GM/r = invariant. We know there
is no absolute force in GR therefore F_u=0.
Then the L,u = (&L/&x^u) =0.
Naturally L,u = L;u. if L is a scalar.
((an exception may occur if L is a density))
>This can equivalently be defined through generalized 4-momentum P_u defined
>as
>
>P_u = &L/&U^u
This cannot work without densities. The LHS is
expressed as a tensor, but the RHS is not a tensor,
because of &U^u. There may be a chance to make
this work if L a density.
Very good analysis and post PmB.
Tucker finds f_u =0, because invariant
accelaration is impossible, and this follows
from invariant acceleration =0.
Can you find an invariant acceleration <>0
in all FoR's? (If so then I'm wrong).
Most of the experienced GRist's would
agree with you, when you say f_u <>0,
especially when Lorentz forces are exhibited.
It is a Haloed moment when Pmb, Bilge and
TR agree tucker is wrong, ((I'll drink to that)).
Essentially what I do is to convert Pmb's force,
f^u = f*U^u, then find the invariant
f=0 and thus f^u=0 always.
In accord with GR, (I'm hard on GR).
Regards
Ken S. Tucker
Pmb
news:2202379a.03121...@posting.google.com...
> "Pmb" <some...@somewhere.com> wrote in message
news:<3OZCb.3610$D66...@nwrdny03.gnilink.net>...
>
> >It is often said that gravity is not a force in GR. That, of course, is a
> >matter of opinion only. Its is entirely based on what one calls "force."
> >
> >For a particle only acted upon by a gravitational force, the 4-force
> >vanishes. The 3-force does not.
> >
> >However there are other ways to GR than 4-vectors. There is Lagrangian
> >mechanics. In Lagrangian mechanics there is the generalized (aka
canonical)
> >4-force F_u defined either as (let & = partial derivative sign)
> >
> >F_u = &L/&x^u
>
> That's a good example, where F_u =0 in GR
You're confusing F_u with the 4-force. These two are not the same animals.
F_u is the generalized force. Not the 4-force
> when L = GM/r = invariant.
Actually that is incorrect. The invariant L is proportional to the rest
energy of the particle
> >P_u = &L/&U^u
>
> This cannot work without densities. The LHS is
> expressed as a tensor, but the RHS is not a tensor,
Sorry. I made an error. That should have been
P_u = &L/&x^u
Thank you for catching that Ken
> Very good analysis and post PmB.
> Tucker finds f_u =0, because invariant
> accelaration is impossible, and this follows
> from invariant acceleration =0.
Notice how the generalized force, F_u, is defined. It's not DP^u/dT which is
a tensor, it's defined as dP_u/dT which is *not* a tensor. It can't be since
the gravitational force has a relative existance.
When you think "gravitational force" please think "Coriolis force" and then
you'll keep straight in your mind that 4-force and inertial force are very
different animals.
The quantity F_u is given by
F_u = (1/2) m_o g_ab,u U^a U^b
For proof see
http://www.geocities.com/physics_world/gr/grav_force.htm
Please see "Gravitation and Spacetime - 2nd Ed.," Hans C. Ohanian, Remo
Ruffini for the derivation. See page 151, Eq. (76) and page 178.
I'm updating the above web site to include this. The first part (which is
all that there is there at the moment) derives this in one way. Another
equivalent way uses the Lagrangian in a more direct way.
There is yet another similar way to ge this result. I did it to derive
conservation laws. See
http://www.geocities.com/physics_world/gr/conserved_quantities.htm
Eq. (4) is the gravitational force too.
Pmb
Pmb
news:<3OZCb.3610$D66...@nwrdny03.gnilink.net>...
>
> >It is often said that gravity is not a force in GR. That, of course, is a
> >matter of opinion only. Its is entirely based on what one calls "force."
> >
> >For a particle only acted upon by a gravitational force, the 4-force
> >vanishes. The 3-force does not.
> >
> >However there are other ways to GR than 4-vectors. There is Lagrangian
> >mechanics. In Lagrangian mechanics there is the generalized (aka
canonical)
> >4-force F_u defined either as (let & = partial derivative sign)
> >
> >F_u = &L/&x^u
This is rather confusing. I don't understand Ohanian's derivation now that I
see what you're saying. I thought that the quantity P_u as he defines it was
the covariant momentum. I don't see any error in his text.
Do you have Ohanian's text?
The actuall calculation is trivial now that I tried it. The Lagrangian for a
particle in free-fall can be expressed as
L = (1/2)m_o g_ab U^a U^b
&L/&x^u = (1/2)m_o g_ab,u U^a U^b
This is the gravitational force. I'll use G_u so as to minimize confusion
with 4-force. Thus
G_u = (1/2)m_o g_ab,u U^a U^b
This is not supposed to be a tensor and yet it sure seems like one to me.
What's wrong with it?
Pmb
Bill Hobba
> It is often said that gravity is not a force in GR. That, of course, is a
> matter of opinion only. Its is entirely based on what one calls "force."
It is based on the lessons learnt from the principle of general invariance
which says the laws of nature should be put in a form that is the same in
all coordinate systems (the principle of general covariance) and when in
that form terms can be divided into absolute and dynamical variables. Since
gravitational 'forces' can be made to disappear locally it is not really a
concept that fits easily into the GR paradigm.
Thanks
Bill
Pmb
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote
[pmb wrote]
Ken - I now have a handle on what's going on. I was reading chapter 3 of
"Gravitation and Spacetime - 2nd Ed." Ohanian and Ruffini. In that section
the authors treat the background spacetime as flat and treat gravitation as
a field in that background!
In that context the derivative with respect to Tau of tensor is a tensor. In
this context the gravitational force is dP_u/dT = (1/2) g_ab,u U^a U^b is a
4-vector. However that relation is exact in the exact solution for dP_u/dT
in a curved spacetime as I've shown on my web site.
So when Ohanian and Ruffini refer to the gravitational force in chapter 3
they consider it a 4-force!
Pmb
car...@no-dirac-spam.ucdavis.edu
> It is often said that gravity is not a force in GR. That, of course, is a
> matter of opinion only. Its is entirely based on what one calls "force."
That's true, and a lot of argument over this comes from different
definitions. On the other hand, if you want to communicate with
people who actually work in GR, it's a good idea to use what is
now the standard definition among professionals, which is the
4-force; or, if you don't, to preface your remarks by stating
explicitly that you're not using that definition.
(If your interest is to coimmunicate with historians of science,
you might choose differently, though there the definitions
are so nonstandard that it's probably a good idea to explain
what definition you're using no matter what it is.)
> For a particle only acted upon by a gravitational force, the
> 4-force vanishes.
True (up to your definition of ``gravitational force,'' which
threatens to make this whole argument circular).
> The 3-force does not.
That depends. With some choices of coordinates, the 3-force
does not vanish. With others it does. Note that the same is
true for a particle acted on by any force whatsoever, or no force
at all -- you can always find some frame in which the 3-force
on a particular particle vanishes, and others in which it does
not.
(This is the reason professionals in the field tend to not define
force as 3-force -- for such a quantity to have physical meaning,
you have to specify both its value and the coordinates you've
chosen.)
> However there are other ways to GR than 4-vectors. There is
> Lagrangian mechanics.
The Lagrangian approach to GR involves 4-vectors.
> In Lagrangian mechanics there is the generalized (aka canonical)
> 4-force F_u defined either as (let & = partial derivative sign)
> F_u = &L/&x^u
Note that this does *not* give something proportional to
the Christoffel connection. (Do the math.) Note also that
if you take the Lagrangian for a charged particle in an
electromagnetic field, it does not give the Lorentz force.
The connection and the Lorentz force are there in the
final equations of motion, but they come from a mixtur
of the term you want to call ``force'' and the term you
want to call ``derivative of the momentum.''
I would suggest that a definition of ``force'' that does not
correctly give the Lorentz force on a charged particle is
not very useful.
Steve Carlip
Pmb
<car...@no-dirac-spam.ucdavis.edu> wrote in message
news:brl06u$d00$2...@woodrow.ucdavis.edu...
> In sci.physics.relativity Pmb <some...@somewhere.com> wrote:
> > It is often said that gravity is not a force in GR. That, of
> > course, is a matter of opinion only. Its is entirely based on what
> > one calls "force."
> That's true, and a lot of argument over this comes from different
> definitions. On the other hand, if you want to communicate with
> people who actually work in GR, it's a good idea to use what is
> now the standard definition among professionals, which is the
> 4-force;
This thread was started with the idea in mind that different relativists
mean different things.
Almost all professionals that I know of (or have read their papers) say
"force" when then mean dp/dt and they say "4-force" when they mean dP/dT (P
= 4-momentum)
For example:
(1) MTW use the term "gravitational force" to mean the Christoffel symbols.
(2) Thorne and Blanchard say "4-force" when they mean 4-force in their new
text
(http://www.pma.caltech.edu/Courses/ph136/yr2002/chap01/0201.2.pdf)
(2) When Ohanian/Ruffini and Cliff Will state that the gravitational force
is velocity dependant they obviously don't mean 4-velocity
(3) Jackson refers to dp/dt = q[E + vxB] as the "Lorentz force equation" and
refers to the 4-force equation as the covariant form
In short - From what I see it is quite clear that one can't assume that the
term "force" when it appears in mondern relativity literature always means
"4-force" or "3-force".
My personal opinion is that if one means 3-force then they should say
"force" and if they mean 4-force then they should say "4-force." Same with
momentum; if someone means 3-momentum then they should say "momentum" and if
they mean 4-momentum then they should say "4-momentum." It makes it easier
to explain what the components are to students who are learning it.
Alternate rule (not something I'd use myself)
If someone means 3-force then they should say "3-force" and if they
mean 4-force then they should say "force". If someone means
3-momentum then they should say "3momentum" and if they mean
4-momentum then they should say "momentum."
> > The 3-force does not.
> That depends. ...
Yes. I agree - Depends on the frame of referance since the gravitational
force can be transformed away. That's the nature of the equivalence
principle of course.
> (This is the reason professionals in the field tend to not define
> force as 3-force -- for such a quantity to have physical meaning,
> you have to specify both its value and the coordinates you've
> chosen.)
Same in Newtonian mechanics when it comes to momentum, energy etc.
> > However there are other ways to GR than 4-vectors. There is
> > Lagrangian mechanics.
> The Lagrangian approach to GR involves 4-vectors.
> > In Lagrangian mechanics there is the generalized (aka canonical)
> > 4-force F_u defined either as (let & = partial derivative sign)
> > F_u = &L/&x^u
> Note that this does *not* give something proportional to
> the Christoffel connection.
I noted this to Ken Tucker ealier this morning (or was it last night). I
made an error. I was thinking of something else. I was reviewing Ohanian's
text. I wrote this thread as a result. In the section I was reviewin they
treat gravity as a field in a flat spacetime background. As such the 4-force
is dP_/dT. I forgot that point.
> Note also that
> if you take the Lagrangian for a charged particle in an
> electromagnetic field, it does not give the Lorentz force.
It gives the canonical force.
> I would suggest that a definition of ``force'' that does not
> correctly give the Lorentz force on a charged particle is
> not very useful.
I believe that people should always be clear on these things. Most
usually are when they publish something. The problem you just
mentioned gets me irritated too. In classical mechanics I find it
irritating that some people say "momentum" when they really mean
"canonical momentum".
Thanks
Pete
Bilge
I tried to find a way to say something similar to what you posted,
but could not come up with a way to say that as well as you did here.
That was really a nice, concise description.
Ken S. Tucker
>"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
>news:2202379a.03121...@posting.google.com...
>> "Pmb" <some...@somewhere.com> wrote in message news:<3OZCb.3610$D66...@nwrdny03.gnilink.net>...
>>
>> >It is often said that gravity is not a force in GR. That, of course, is a
>> >matter of opinion only. Its is entirely based on what one calls "force."
>> >
>> >For a particle only acted upon by a gravitational force, the 4-force
>> >vanishes. The 3-force does not.
>> >
>> >However there are other ways to GR than 4-vectors. There is Lagrangian
>> >mechanics. In Lagrangian mechanics there is the generalized (aka canonical)
>> >4-force F_u defined either as (let & = partial derivative sign)
>> >
>> >F_u = &L/&x^u
>>
>> That's a good example, where F_u =0 in GR
...
Ok, however F_u =&L/&x^u = L,u =tensor,
when L is invariant.
> That should have been
>
>P_u = &L/&x^u
Ok, I presume P_u = F_u...
>> Tucker finds f_u =0, because invariant
>> accelaration is impossible, and this follows
>> from invariant acceleration =0.
>
>Notice how the generalized force, F_u, is defined. It's not DP^u/dT which is
>a tensor, it's defined as dP_u/dT which is *not* a tensor. It can't be since
>the gravitational force has a relative existance.
Ok, but far above F_u was in fact a tensor. (L,u)?
>When you think "gravitational force" please think "Coriolis force" and then
>you'll keep straight in your mind that 4-force and inertial force are very
>different animals.
>
>The quantity F_u is given by
>
>F_u = (1/2) m_o g_ab,u U^a U^b
The RHS is not a tensor.
>For proof see
I'll hold here until clarication.
[snip]
>Pmb>
Ken S. Tucker
Igor
> It is often said that gravity is not a force in GR. That, of course, is a
The main problem is that bodies moving on geodesic paths like we have
in GR have no net forces acting on them. Remember that a force is a
vector quantity (whether you express it in 3-space or 3+1 space-time).
The main difference is that geodesic acceleration can vanish in a
particular frame (free-fall), but still be non-zero in other frames.
This is not the property of a true vector, which, if it vanishes in
one frame, must vanish in all frames in order to remain covariant.
Your quote from Einstein embraces Machian concepts, which are usually
taken with a grain of salt by most students of GR these days.
Einstein tried for a long time to fit Mach's concepts into GR and
never succeeded. He finally gave up on Mach toward the end. But
there are still some people who just keep right on trying. Good luck.
Gauge
> Since
> gravitational 'forces' can be made to disappear locally it is not really a
> concept that fits easily into the GR paradigm.
How do you think the Coriolis force fits into Newtonian mechanics?
Pmb
Pmb
Pmb
"Bill Hobba" <bill...@yahoo.com.au> wrote in message
news:3fdd4...@news.iprimus.com.au...
> Pmb wrote:
That is not what the "GR paradigm" is all about. That is part of it - not
the whole thing. In fact you're omitting the very concepts which led
Einstein to create GR and that's Mach's Principle.
Instead of me repeating what I've already explained to others take a look at
"Cosmological Principles," John A. Peacock, Cambridge University Press,
(1999) to see what I'm refering to. I.e. see the section called "Inertial
frames and Mach's Principle." The relavent chapter is online at
http://assets.cambridge.org/0521422701/sample/0521422701WS.pdf
And the fact that the gravitational force/field can be transformed away is
simply the equivalence principle. The inertial frame played an important
role in Newtonian mechanics but it does not hold its priveliged status in
GR. Notice what Peacock concludes in that section
-------------------------
Thus, it does seem qualitatively valid to think of inertial forces as
arising from gravitational radiation. Apart from being a starlingly
different view of what is going on in non-inertial frames, this arguement
also sheds light on Mach's principle: for a symetric universe, inertial
forces clearly vanish in the average rest frame of the matter of the
universe.
-------------------------
But as I said - this is all semantics as far as what the word "force" means.
Pmb
[Moderator's note: followups have been set to sci.physics.relativity,
which is the suitable place for discussions of this sort. - jb]
Danny Ross Lunsford
> It is for the above reasons that some relativists hold that there is a
> gravitational force and that it is a real force.
What do you mean "real"? The gravitational force in GR is very real, it
bends light beams. What you are really saying is that an energy-momentum
tensor for some "gravity field" can be put on the right side in Einstein's
equations.
But what would such a thing consist of? Well, the gravitational field
is given by the g_{mn} by hypothesis. So this energy tensor would have
to be symmetric and formed from the g_{mn} and its derivatives. How
high derivatives? Well up to making it proportional to an energy
tensor, that is, second, and those linear. And it has to be
covariantly conserved. But what, there is only one such tensor, and
it's the Einstein tensor. So all I really did was plop down a copy of
the Einstein tensor on the right side with some arbitrary constant. I
can pull this over to the left and all I've done is change my
definition of the gravitational "constant" - so the existence of such
a field is has no separate meaning. All you really did was demonstrate
that GR has no background.
--
-drl
Doug Sweetser
You have an equation for generating the force:
> F_u = &L/&x^u
There are several choices one could make for the Lagrange density, one
being R (-g)^(1/2), another being the Palatini method. A search of the
literature should be able to turn up someone else toiling in this soil.
I prefer to try and do calculations like this myself, having a hard
time finding the relevant equation in a book or article.
Personally, I see no connection to Mach's principle because the
equation above looks like a straight-forward calculation.
doug
quaternions.com
[Moderator's note: followups have been set to sci.physics.relativity. - jb]
Pmb
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
news:2202379a.03121...@posting.google.com...
> "Pmb" <some...@somewhere.com> wrote in message
news:<3OZCb.3610$D66...@nwrdny03.gnilink.net>...
> >It is often said that gravity is not a force in GR. That, of course, is a
> >matter of opinion only. Its is entirely based on what one calls "force."
> >
> >For a particle only acted upon by a gravitational force, the 4-force
> >vanishes. The 3-force does not.
> >
> >However there are other ways to GR than 4-vectors. There is Lagrangian
> >mechanics. In Lagrangian mechanics there is the generalized (aka canonical)
> >4-force F_u defined either as (let & = partial derivative sign)
> >
> >F_u = &L/&x^u
> That's a good example, where F_u =0 in GR
> when L = GM/r = invariant. We know there
> is no absolute force in GR therefore F_u=0.
> Then the L,u = (&L/&x^u) =0.
>
> Naturally L,u = L;u. if L is a scalar.
>
> ((an exception may occur if L is a density))
> >This can equivalently be defined through generalized 4-momentum P_u
> >defined as
> >
> >P_u = &L/&U^u
> This cannot work without densities. The LHS is
> expressed as a tensor, but the RHS is not a tensor,
> because of &U^u. There may be a chance to make
> this work if L a density.
I incorrectly corrected myself! :-)
The expression
P_u = &L/&U^u
is correct.
More later
Pmb
Gauge
news:<brhu0d$ks4$1...@pcls4.std.com>...
> You have an equation for generating the force:
>
> > F_u = &L/&x^u
>
> There are several choices one could make for the Lagrange density, one
> being R (-g)^(1/2),
I was not talking about the Lagrangian density. That is the Lagrangian
for a field. I was refering to the Lagrangian of a particle moving in
a field.
However I neglected to say that I was thinking of something else -
this is a treatment using a flat spacetime background and treating
gravity as a field in that spacetime.
Bilge
>news:<brhu0d$ks4$1...@pcls4.std.com>...
>
>> You have an equation for generating the force:
>>
>> > F_u = &L/&x^u
>>
>> There are several choices one could make for the Lagrange density, one
>> being R (-g)^(1/2),
>
>I was not talking about the Lagrangian density. That is the Lagrangian
>for a field. I was refering to the Lagrangian of a particle moving in
>a field.
>
>However I neglected to say that I was thinking of something else -
>this is a treatment using a flat spacetime background and treating
>gravity as a field in that spacetime.
>
You meam like newtonian mechanics?
Doug Sweetser
> >I was not talking about the Lagrangian density. That is the
> >Lagrangian for a field. I was referring to the Lagrangian of a
> >particle moving in a field.
>
> Then you aren't really talking about a relativistic formulation.
This might not be unreasonable. Think about EM. If one want to
concentrate on the field, one uses this Lagrange density:
L = - J_u A^u/c - 1/4 c^2 (A^u,v - A^V,u)(A_u,v - A^v,u)
Take the volume integral of L to get the action. By varying the action
with A^u and A^u,v, one gets the Maxwell field equations.
If one wants to think about a particle moving in a field, then this
Lagrange density is relevant:
L = - rho/gamma - J_u A^u/c
where
rho is the mass charge density
gamma is 1/(1 - v.v)^(1/2)
Form the action in the usual way. By taking the variation with respect
to the velocity v, the Lorentz 4-force is generated. That is how EM
can deal with a particle moving in a field. [I was once criticized for
having all three terms in the same Lagrange density, which struck me as
odd because what matters in generating equations from a Lagrange
density is the choice of variables to do the variation, A^u and A^u;v
to get the field equations, v to get the 4-force.]
It does not appear like a similar trick will work for general
relativity. In EM, both gamma and J_u contain a velocity v, so one can
do the calculus of variations with respect to v. The Lagrange density
of general relativity is exclusively about metrics and the changes in
metrics, not velocities.
doug
quaternions.com
Gauge
> Hello Bilge:
>
> > >I was not talking about the Lagrangian density. That is the
> > >Lagrangian for a field. I was referring to the Lagrangian of a
> > >particle moving in a field.
> >
> > Then you aren't really talking about a relativistic formulation.
>
> This might not be unreasonable. Think about EM. If one want to
> concentrate on the field, one uses this Lagrange density:
>
> L = - J_u A^u/c - 1/4 c^2 (A^u,v - A^V,u)(A_u,v - A^v,u)
>
> Take the volume integral of L to get the action. By varying the action
> with A^u and A^u,v, one gets the Maxwell field equations.
>
> If one wants to think about a particle moving in a field, then this
> Lagrange density is relevant:
>
> L = - rho/gamma - J_u A^u/c
Where did this come from? Are you saying that when you plug this into
Lagrange's equations you get out the Lorentz force equation? After all
that is the equation for a particle moving in a field
>
> where
> rho is the mass charge density
> gamma is 1/(1 - v.v)^(1/2)
>
> Form the action in the usual way. By taking the variation with respect
> to the velocity v, the Lorentz 4-force is generated. That is how EM
> can deal with a particle moving in a field. [I was once criticized for
> having all three terms in the same Lagrange density, which struck me as
> odd because what matters in generating equations from a Lagrange
> density is the choice of variables to do the variation, A^u and A^u;v
> to get the field equations, v to get the 4-force.]
>
> It does not appear like a similar trick will work for general
> relativity. In EM, both gamma and J_u contain a velocity v, so one can
> do the calculus of variations with respect to v. The Lagrange density
> of general relativity is exclusively about metrics and the changes in
> metrics, not velocities.
For a charged particle moving in an EM field a (relativistic,
non-covariant) Lagrangian is given by
L = -m_o*sqrt[1-(v/c)^2] - q*Phi - (q/c)v*A
For two covariant Lagrangian's for a charged particle in an EM field
see Eq. (28) and Eq. (35) in
http://www.geocities.com/physics_world/relativistic_charge.htm
Pmb
Bill Hobba
'Since gravitational 'forces' can be made to disappear locally it is not
PMB replied:
> How do you think the Coriolis force fits into Newtonian mechanics?
Errrr Acceleration wrt to an inertial frame. Newtonian mechanics does not
have the law of general invariance - it deals with inertial reference frames
so it only has the POR. Accelerated reference frames are handled by
transforming to an inertial reference frame (exactly like GR but the law of
general covariance is neither required or assumed) and assuming that time is
absolute (which is the key assumption that allows us to proceed).. The
lagrangian formalism does not even introduce the concept of force in such
situations. See page 126 - Landau - Mechanics for a discussion on this and
the introduction of the centrifugal potential energy to mathematically
handle such situations.
Thanks
Bill
Gauge
> Hello Pmb:
>
> You have an equation for generating the force:
>
> > F_u = &L/&x^u
>
> There are several choices one could make for the Lagrange density, one
> being R (-g)^(1/2), another being the Palatini method. A search of the
> literature should be able to turn up someone else toiling in this soil.
> I prefer to try and do calculations like this myself, having a hard
> time finding the relevant equation in a book or article.
>
> Personally, I see no connection to Mach's principle because the
> equation above looks like a straight-forward calculation.
Mach's Principle corresponds to inertial forces. When DP_u/DT = 0 and
dP_u/dT is not zero then dP_u/dT is a pure inertial force.
As far as Mach's Principle - This concept is found in almost all GR
texts including Ohanian, Wald and D'Inverno. Its the same arguements
in these newer text that Einstein gave all those years ago.
Mach's view is states that inertial forces are generated by all other
matter in the universe.
Consider a spherical shell at rest in an inertial frame. Inside the
shell the gravitational force on a particle is zero (Think of the
shell as "the distant stars"). Now start rotating the shell. A
particle inside the shell will experience an inertial force. These are
just the centripetal and Coriolis forces.
Frame dragging is an example consistent with Mach's principle.
Wheeler phrased Mach's Principle in 1964 as follows
---------------------------
The inertial properties of an object are determined by energy-momentum
throughout all space.
---------------------------
Pmb
Ken S. Tucker
Well, Sometimes, g_ij = d_ij + h_ij where
d_ij = +/-1 or 0 as i=j or i=/=j, I think this is
referred to as the weak-field solution.
In this sense, h_ij is a field imposed on the
flat (euclidian) background d_ij.
In view of Mr. Sweeter's remarks, the det|g_uv|
in t,x is
g = | 1+h_00 h_01 | = -1 - (h_01)^2
| h_10 -1-h_11 |
for example (h_01 is symmetric) and
(1+h_00) *(-1-h_11) = -1.
Then g becomes a true *relativistic* variable.
Setting L =R (-g)^(1/2) and setting R to an
*absolute* invariant means L is a *relative*
invariant. (See Dover's P of R, "Hamiltons
Principle" pg 167, for the use of relative
invariants, as I understand).
Ken S. Tucker
PS: Snippable intrepretation...
IMO if L=R then the Einstein Effect vanishes,
because g = constant.
OTOH, the "h_01" term is the Escape velocity
at any point, ie.
V(e)^2 = 2GM/r. (for example),
and L becomes a relative invariant.
Then L,u is meanigless, and needs to be replaced
by the covariant derivative L;u =0.
That's standard, when going to GR from the
ordinary derivatives and are replaced by covariant
derivatives.
KST
car...@no-dirac-spam.ucdavis.edu
> <car...@no-dirac-spam.ucdavis.edu> wrote in message
> news:brl06u$d00$2...@woodrow.ucdavis.edu...
>> In sci.physics.relativity Pmb <some...@somewhere.com> wrote:
>> > It is often said that gravity is not a force in GR. That, of
>> > course, is a matter of opinion only. Its is entirely based on what
>> > one calls "force."
>> That's true, and a lot of argument over this comes from different
>> definitions. On the other hand, if you want to communicate with
>> people who actually work in GR, it's a good idea to use what is
>> now the standard definition among professionals, which is the
>> 4-force;
> This thread was started with the idea in mind that different relativists
> mean different things.
> Almost all professionals that I know of (or have read their papers) say
> "force" when then mean dp/dt and they say "4-force" when they mean
> dP/dT (P= 4-momentum)
The examples you give are all introductory textbooks, in which the
authors sometimes use the terminology their students will be more
familiar with from introductory pre-relativity courses. If you look
at published papers are listen to talks at conferences, you will find
that ``force'' virtually always means ``4-force'' unless explicitly
stated otherwise.
You are, of course, free to believe me or not.
Steve Carlip
Pmb
> Bill Hobba wrote:
> 'Since gravitational 'forces' can be made to disappear locally it is not
> really a concept that fits easily into the GR paradigm.'
>
> PMB replied:
> > How do you think the Coriolis force fits into Newtonian mechanics?
>
> Errrr Acceleration wrt to an inertial frame.
So you admit that it "fits in" (whatever that means) to Newtonian mechanics.
Since the situation is no different in GR why would you think it doesn't
"fit in" to GR?
> So why do you think its different in GR? Newtonian mechanics does not
> have the law of general invariance - it deals with inertial reference
frames
> so it only has the POR.
Newtonian mechanics does not deal with just inertial frames. Where would you
get an idea like Like that?
> Accelerated reference frames are handled by
> transforming to an inertial reference frame (exactly like GR but the law
of
> general covariance is neither required or assumed) and assuming that time
is
> absolute (which is the key assumption that allows us to proceed).
One doesn't always do that. In fact it is often wise not to in some cases
> The
> lagrangian formalism does not even introduce the concept of force in such
> situations.
I *highly* disagree with you on that point! That is not true by any means.
The Lagrangian formalism is very well equiped to deal with such situations
> See page 126 - Landau - Mechanics for a discussion on this and
> the introduction of the centrifugal potential energy to mathematically
> handle such situations.
Did you read that entire section Bill? Please turn to page 128 and see the
results of their efforts. After Eq. (39.7) the authors write
**********************************
We see that the "inertial forces" due to rotation of the frame consists of
three terms. The force mr x Omega' is due to the non-uniformity of rotation,
but the other two terms appear even if the rotation is uniform. The force
2mv x Omega is called the Coriolis force; unlike any other (non-dissipative)
force hitherto considered, it depends on the velocity of the particle. The
force m*Omega x (r x Omega) is called the centrifugal force. [etc].
**********************************
That is a perfectly standard derivation which in no way is different than
any other derivation using generalized potentials. In fact its quite like
the Lagrangian for the Lorentz force in every way possible. The term mr x
Omega' is refered to as the Euler-Force (a term coined by Lanczos)
Regarding the "centrifugal potential." What's your point? You seem to be
thinking that this is some sort of mathematical trickery to get things to
come out pretty. It is not so my any means. This is standard interpretation.
There are two potentials used in Lagrangian mechanics. Label them V and U.
The former is potential energy and the later the generalized potential. They
are related by
V = q'_i &U/&q'_i - U
Try this out for yourself and see. Let U be the generalized potential for a
charged particle moving in an EM field. Then you'll see that V = q*Phi is
the potential energy of the charged particle where Phi = Coulomb potential.
The energy function h will then be the energy and will have the value
E = T + V
Pmb
Pmb
Another comment - I noticed that you prefer Lagrangian formulations. Did you
read that page in Landau on non-inertial frames and inertial forces. On page
126 they write
**************************************************
Let us now consider what the equations of motion will be in an inertial
frame of referance. The basis of the solution of this problem is again the
principle of least action, whose validity does not depend on the frame of
referance chosen.
**************************************************
On page 127 they write
**************************************************
Thus an accelerated translational motion of a frame of referance is
equivalent, as regards to mechanical effects on the equations of motion of a
particle, to the application of a uniform field of force equal tot the mass
of the particle multiplied by the acceleration W, in the direction opposite
to this acceleration.
**************************************************
Compare this to Einstein's remarks in his 1916 GR paper
**************************************************
Let K be a Galilean system of reference, i.e. a system relatively to which
(at least in the four-dimensional region under consideration) a mass,
sufficiently distant from other masses, is moving with uniform motion in a
straight line. Let K' be a second system of reference which is moving
relatively to K in uniformly accelerated translation. Then, relatively to
K', a mass sufficiently distant from other masses would have an accelerated
motion such that its acceleration and direction of acceleration are
independent of the material composition and physical state of the mass.
Does this permit an observer at rest relatively to K' to infer that he is on
a "really" accelerated system of reference? The answer is in the negative;
for the above-mentioned relation of freely movable masses to K' may be
interpreted equally well in the following way. The system of reference K' is
unaccelerated, but the space-time territory in question is under the sway of
a gravitational field, which generates the accelerated motion of the bodies
relatively to K'.
This view is made possible for us by the teaching of experience as to the
existence of a field of force, namely, the gravitational field, which
possesses the remarkable property of imparting the same acceleration to all
bodies.
**************************************************
Zero difference. Notice what Einstein is refering to when he says "The
answer is in the negative" - I.e. that one is not to infer that he is on a
"really" accelerated system.
Pmb
Pmb
----- Original Message -----
From: "Bill Hobba" <bill...@yahoo.com.au>
Newsgroups: sci.physics.relativity,sci.physics.research
Sent: Tuesday, December 23, 2003 3:38 PM
Subject: Re: Gravitational force and Lagrangian Mechanics
Another comment - I noticed that you prefer Lagrangian formulations. Did you
Pmb
<car...@no-dirac-spam.ucdavis.edu> wrote in message
news:brtcq0$6r6$2...@woodrow.ucdavis.edu...
First off I don't consider Ohanian an introductory text. And I don't get
that same impression that you do as I read the literature either since I too
am refering to the same literature. In fact one comes to mind - A paper by
Thorne, Braginski and Caves refers to the gravitational force in one paper I
read. It was that paper regarding gravitomagnetism. There were also papers
in the american journal of physics too and there were those papers on mass
by Lev Okun where Okun argues his point using the relations on gravitational
force I've been speaking about.
Referances furnished upon request. Of course you too are free to believe me
or not.
Pmb
car...@no-dirac-spam.ucdavis.edu
> <car...@no-dirac-spam.ucdavis.edu> wrote in message
> news:brtcq0$6r6$2...@woodrow.ucdavis.edu...
[...]
>> The examples you give are all introductory textbooks, in which the
>> authors sometimes use the terminology their students will be more
>> familiar with from introductory pre-relativity courses. If you look
>> at published papers are listen to talks at conferences, you will find
>> that ``force'' virtually always means ``4-force'' unless explicitly
>> stated otherwise.
> First off I don't consider Ohanian an introductory text.
Of course it is. It's a textbook for a first course in general relativity,
written in a way designed, in the words of the preeface, ``for students
in their first encounter with general relativity.''
> And I don't get that same impression that you do as I read the
> literature either since I too am refering to the same literature.
Well, in the past two years I've written seven papers, organized one
conference and spoken at five (low because I have a new baby),
served on the editrial board of the leading specialized journal in
general relativity, and refereed about 75 papers. This gives me a
pretty good basis for my ``impression,'' I think.
> In fact one comes to mind - A paper by Thorne, Braginski and
> Caves refers to the gravitational force in one paper I read. It
> was that paper regarding gravitomagnetism.
This is a paper on experimental tests in the post-Newtonian
approximation, in which there is a clear preferred reference
frame (the lab) and a clear Newtonian limit that defines what
one might mean by ``force.'' It's not typical.
> There were also papers in the american journal of physics
The AJP is not a technical journal. It is designed to publish
`` papers that meet the needs and intellectual interests of college
and university physics teachers and students'' that ``significantly
aid the process of learning physics.'' Its Statement of Editorial
Policy, from whhich I've taken these quotes, goes on to say,
``Manuscripts announcing new theoretical or experimental
results or manuscripts questioning well-established and
successful theories are not acceptable and should be submitted
to an archival research journal for evaluation by specialists.''
I enjoy reading the AJP, and have even published a couple of
papers there. But it's not the place professionals go to talk
about or learn about general relativity -- it's, at best, a place
we go to learn about nice ways to *teach* general relativity,
which is an entirely different matter.
Steve Carlip
Pmb
<car...@no-dirac-spam.ucdavis.edu> wrote
> The AJP is not a technical journal. It is designed to publish
> `` papers that meet the needs and intellectual interests of college
> and university physics teachers and students'' that ``significantly
> aid the process of learning physics.'' Its Statement of Editorial
> Policy, from whhich I've taken these quotes, goes on to say,
It also has served to guide understanding of the basic principles of
physics. One would publish a paper in that journal if they have something to
say about the fundamentals of physics.
I'm unclear as to why a certain journal is somehow unworthy of being a
physics journal since all I said was that some jorunals do speak of the
gravitational force as I've mentioned. And this is true. That journal is for
teaching the best ways to think. And if something appears in it about force
then its there to teach the best way to think about force. E.g. Edwin F.
Taylor wrote a guest paper which appeared in that journal. The brunt of the
paper was to dispense with the concept of force. And this was a basic law of
Newtonian physics that we're talking about. However as time marches on its
becoming clear that force is not required in mechanics other than to express
general statements and to deal with things like friction. However one never
has to even mention force when dealing with monogenic systems.
I'm also unclear as to why you think that because I said that "force"
appears in a gravitational context that it doesn't qualify in some cases
because of the reasons you gave. What does "typical" have to do with this?
If I gave you the impression that 99% of all GR papers (or something like
that) use the notion of g-force then you go the wrong impression. I'm
speaking of a wide variety of journals (and articles published over many
years). In reading so many different journals and since I was
**researching** "gravitational force" for so long I was keenly aware of such
usage and the articles in which it is discussed. Consider how you select
articles and which articles you do read. Then it may become clear since when
different people have different interests in the same field they will read
different things.
A friend of mine is/was a ref for that journal. He is a well-known GR
physicists. He worked in the field for many many many years (even before I
was born). He's a *very* sharp man. And in all that time he was never aware
that, in Newtonian gravity, there was a finite distribution of matter which
gave rise to a perfectly uniform g-field. That just goes to show you that
we're all human and we're all unware of basic things even in our own fields
of expertise.
If you ref so many papers and saw the notion used more than a fraction of
the time (if at all) then I'd be surprised. However that has little to do
with the notion itself.
Suppose you ref'd 75 papers and gravitational force appears only in 1% of
them. The odds are less than 50/50 that you'd read it.
For example: Its well known that the wheel works. But an article won't come
across your desk which describes the operation of the wheel either. And that
would include all mechanics papers. Typically its not the subject.
But I don't see why AJP is less worth of discussion that a GR journal. Why
does it matter to you since all I said was that the notion of gravitaitonal
force in GR appears in the physics literature, physics texts and journals
As far as texts - Consider the purpose of a text. It is created to teach a
student a subject and to teach him how to think. Texts are always
introductory in that sense. Ohanian is no less a GR text than Wald. When you
get to something like Wald you've learned it and you're not really being
taught GR. You're being brought up to speed on the hard core mathematics
etc.
> ``Manuscripts announcing new theoretical or experimental
> results or manuscripts questioning well-established and
> successful theories are not acceptable and should be submitted
> to an archival research journal for evaluation by specialists.''
That is not quite accurate. It gives the impression that fundamental ideas
are not addressed. E.g. it may not be the place to question theories but it
is the place to question concepts. E.g. you don't use it to question
classical mechancis. But you do use it to question the use of force in
classical mechanics.
If you ref'd papers in GR then I also wouldn't imagine that the notion came
up that much. Again - you may have got the impression that I said that it
appears 99% of the time instead of 1% of the time. As such I'd be surprised
if you read anything like that.
However I had a conversation with my GR prof on this. When I explained to
him what I was doing and explained the idea I had which fell out of the
concept of gravitational force he was quite surprised and delighted.
Especially since it was so simple and easy to see once you've though about
gravitational force a bit.
> I enjoy reading the AJP, and have even published a couple of
> papers there. But it's not the place professionals go to talk
> about or learn about general relativity -- it's, at best, a place
> we go to learn about nice ways to *teach* general relativity,
> which is an entirely different matter.
Not entirely. Teaching and thinking about it should be closely related. For
example: When you get a paper to ref all you get is a sterilized version. It
does not tell you what the author was thinking when he wrote it nor does it
tell you what thinking led him down the wrong paths as he was researching it
and what led him down the correct paths.
Have you ever spent a lot of time teaching a student a way of thinking only
to end up telling him not to think that way?
Thanks for your thoughts.
Pmb
Ken S. Tucker
[snip about AJP for now, more L8R]
>However I had a conversation with my GR prof on this. When I explained to
>him what I was doing and explained the idea I had which fell out of the
>concept of gravitational force he was quite surprised and delighted.
>Especially since it was so simple and easy to see once you've thought about
>gravitational force a bit.
Not simple for me. Let's define *force* as a non-zero
reading on an accelometer.
In every instance this requires the intervention of
Electromagnetic and/or Quantized forces. For examples
1) An accelerating Space-ship subject to thruster action
has a momentum reaction that is ultimately dependant on
the electromagnetic repulsion of the exhaust gas molecules
relative to the combustion chamber wall. These momentum
reactions ultimately exchange quanta (photons) in this
reaction.
(this requires an expenditure of energy).
2) We're sitting on a chair sensing force that has an
electromagnetic origin in the electro-mechanical stress
in the chair, created by our weight.
(this does not require an expenditure of energy).
3) It seems clear the non-zero accelometer reading
cannot vanish by CS transformation. (Should add
GR certainly allows accelerating CS's to regard
themselves at rest, as we are at our desks now,
ie. you can be the center of the universe).
Therefore the non-zero reading on the
accelometer is invariant, (although the actual
quantity may be relative, it won't vanish).
4) The invariant nature of the accelometer reading
is independant of the energy expenditure as (1)
and (2) show.
Summarizing: please see...
Edward G. Harris's, " Analogy between general
relativity and electromagnetism for slowly moving
particles in weak graviatational fields"
Am, J. Phys. 59(5), May 1991.
with special consideration to his eq.s (4a) and (4b).
(I have personal reservations about his definition
of the anti-symmetrical Christoffel, but that's just
math).
Please note from my point's above the need for an invariant
input - to produce a nonzero accelometer reading - based
on electromagnetic effects. Harris has produced this from
the Chrisoffel tensor, though it lacks the rigor GRist typically
desire.
Regards
Ken S. Tucker
car...@no-dirac-spam.ucdavis.edu
In sci.physics.relativity Pmb <some...@somewhere.com> wrote:
> <car...@no-dirac-spam.ucdavis.edu> wrote
> I'm unclear as to why a certain journal is somehow unworthy of being a
> physics journal since all I said was that some jorunals do speak of the
> gravitational force as I've mentioned. And this is true.
I got into this with a statement about the way almost all professionals
use the terminology. I stand by that. If you are now saying that you
just meant that a few professionals use ``force'' the way you describe,
and in a few special contexts, I have no argument with that. But my
point stands about communicating with most experts.
>> ``Manuscripts announcing new theoretical or experimental
>> results or manuscripts questioning well-established and
>> successful theories are not acceptable and should be submitted
>> to an archival research journal for evaluation by specialists.''
> That is not quite accurate.
It's a direct quote from the AJP Statement of Editorial Policy,
(http://www.kzoo.edu/ajp/docs/edpolicy.html), paragraph 5.
> Have you ever spent a lot of time teaching a student a way of
> thinking only to end up telling him not to think that way?
Of course. I imagine every professor who's ever taught an
introductory course has. Students start with simplified,
idealized examples, presented with analogies to what they
already know, and are gradually introduced to more complex
and sophisticated views that are often ultimately very different
from the starting point. For instance, students who first learn
about gravity are taught Newton's force law, with its implied
``action at a distance,'' and spend a lot of time learning to
compute planets' trajectories and the like. Would you prefer
that I start out with differential geometry?
Steve Carlip