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Conservation of momentum is a result of translational symmetry in the Lagrangian.

...

Yep. I'm quite aware of that, thanks. I.e. I've spent a great deal of time working on Lagrangian dynamics, as an under graduate student, in graduate school, and in personal independant studies, e.g.

http://home.comcast....stic_energy.htm as well as Noether's theorem.

It's easy to see: consider a Lagrangian which satisfies the following property:

 

[math]L(x^i,\dot{x}^i,t)=L(x^i+\delta x^i,\dot{x}^i,t)[/math]

 

This means that [math]\partial L / \partial x^i=0[/math], so the Euler-Lagrange equation reduces to:

 

[math]\frac{d}{dt}\left ( \frac{\partial L}{\partial \dot{x}^i} \right )=0[/math]

 

Now, [math]\partial L / \partial \dot{x}^i =p^i[/math] by definition, therefore momentum doesn't change over time.

Yep. I'm also aware of that as well as Noether's theorem which is what you're discussing. That does't tell us anything more than what swansont was trying to say .... unless you thought I didn't understand Lagrangian dynamics and Noether's theorem. In either case, I understand them both. Yay!

Yep. I'm quite aware of that, thanks. I.e. I've sent a great deal of time working on Lagrangian dynamics, as an under graduate student, in graduate school, and in personal independant studies, as in e.g.

http://home.comcast....stic_energy.htm

Yep. I'm also aware of that as well as Noether's theorem which is what you're discussing. That does't tell us anything more than what swansont was trying to say .... unless you thought I didn't understand Lagrangian dynamics and Noether's theorem. In either case, I understand them both. Yay!

Does anybody want to take a crack at figuring out the Lagrangian of a system of three particles, and expressed in terms of each particle's passive gravitational mass and active gravitational mass? Thanks.

Pete

I read about it in science news many years ago while I was looking through old copies of the magazine

Relying on memory in science can be tricky. What the magazine meant by it said and your memory combines to be a problematic article retrieval in your memory. For example: Does the molecule have a spectrum which is the same as an atom etc.

More later

sometimes whole molecules act like giant atoms.

so it would in theory be possible to increase the mass.

but not decrease

Please post such an exmple.

Conservation of momentum is a consequence of the symmetry of of the Lagrangian under a translation in space. Under that condition, it is true by definition.

Establishing a relation by using the Lagrangian, Lagranges equations, and a conservation relationhip is a matter of derivation. It's not true because it was defined that way.

There is something else which bothers me about your arguement which I can't put my finger on. If I figure it out I'll get back to this. I have a feeling that there is trouble forming a Lagrangian. In any case I'm having trouble trying to write down a Lagrangian for a simple ayarwn for, say, a system of three particles moving in the xy-plain. E.g. how does on express the potential terms in tems of acitve and gravitational mass?

Hello, I have some questions about General Relativity that I cant seem to get an answer.

 

I read possible the universe can wrap back in on it self, so if you looked far enough in one direction you could see the back of your head. So this is understood as the space warping in 3 Dimensions.

1- But if the space is really empty & not like a trampoline surface (in 2D), than can you explain how the space doesn't have an edge or boundary if finite in spacial extent? This really bugs me unless I'm missing something obvious?

 

I just don't know how a 3D universe that is curved having no boundary & being finite can be with empty space?

 

thanks

Above you said if the space is really empty. The space can have a density so small that its possible to have a straight line and not run into one of the particles in the space or you can take that one particle out of the way a bit.

What that means is that its possible to have a straight line (a geodesic in 3-d space) from the front of your head to the back of your head but not gonig through your head.

Is it possible to make different atoms of different elements,but with a small mass?Like having an element which behaves like oxygen but it has the mass of hydrogen.

Not to a great extent. What I mean is this - It's possible to have two oxygen oxygen atoms with different mass. This is due to different number of neutrons in the oxygen nucleus.

Something occured to me this morning. I've read/heard people argue about the distinction between active and passive gravitational mass. If these quantities were different then there'd be a violation in conservation of momentum. The conservtion of momentum is based on observation. The equality is therefore a law of physics, and not something which is true by definition.

Let us compare this with the equality of a particle's inertial mass and passive gravitational mass. If this equality were not true then the Equivalence Principle would be wrong. If the equivalence principle was wrong then the rate of a particles rate of fall would depend on the particle's rest mass. Therefore this is also something which is true based on observed. The equality is therefore a law of physics and not something which is true based on definition.

To take not take note of passive g-mass and active g-mass is therefore inconsistent with the taking note of the equality of passive g-mass and inertial g-mass.

this post was a mistake)

Is something technically still existing even when it has 0 energy? I mean I guess if it has 0 energy, it has 0 mass, but if you have something like an "electron field", what is it existing as when it's not oscillating? How would it start some perturbation sequence? Is there some way to force electrons to have the same spin state in the same quantum state and cancel each other out and make them not exist anymore?

I'd hazard to guess that if the components of the stress-energy-momentum tensor of an object is not zero then that object exists. This was Einstein's view in that Eintein that if any of the components of that object do not vanish then there is matter in there is region of space where it is not zero. Are you familiar with tensors etc?

!

Moderator Note

Technically, yes, it is. You could link to this thread from the one you started, saying the ideas are similar. But we frown upon "advertising" speculations in other threads; the cross-pollination is so rarely a good thing.

 

On the funny part of this, there are now 5 posts discussing whether the threads was hijacked.

Sorry, do you mean continuing the discussion of the other thread in this thread would be thread hijacking?

Yes. That is not working.

Just some quick comments on your page: ...

 

Thanks for pointing out the oversites on my part. So many equaions can make me go blind sometimes. Please don't hesitate to point them out in the future. And I thank you vedy vedy much!

As for what you write above, you haven't given a definition of active/passive gravitational mass.

The definition of active and passive gravitational mass are as follows

1) inertial mass m_i is the constant of proportionality between velocity and momentum.

 

2) passive gravitational mass is the mass that gravity acts on

 

3) active gravitational mass is the mass that is the source of gravity

 

It seems like, from what you say about four-momentum, you consider PG-mass to be equal to rest/invariant mass.

Yes. That is correct.

But then you go on to say that invariant mass doesn't equal the magnitude of the four-momentum, so I'm not sure what to make of it.

I don't know where you got that from. Sometime people confuse proper mass with m(0) where m = m(v) = inertial mass = relativistic mass. If a particleis at rest in a gravitational field then then, as always,

|P|/c is the invariant mass. However when v = 0 and the particle is in a g-field (or non-inertial frame of reference) then

|P|/c != m(0) != proper mass

I recommend that you try it out for the most simplest case you can think of. E.g. let the particle be at rest in a uniform g-field or something and see how it works out.

Another way to address this problem is to look at it from another point of view. It's one I don't like but you might enjoy but you might.

First - It seems to me that asking Is gravity really a force or is it something else? starts off similarly to asking Is electricity a force or is it something else? How did we come get to ask these questions? Electricity is a science so how can it come to be that a broad field of study has a field of force to it?

I had a similar thought here:

http://www.sciencefo...__1#entry646544

 

Might be of interest to you.

 

Apologies if this is considered thread hijacking

I believe what you just did i not thread hijacking. I believe that continuing the discussion here is thread hijacking though.

From your link:

 

You must have ment "Eq. (9) is identical to Eq. (3),

I stated Eq. (3) whereupon I decide to prove/derive Eq. (9) which I did. I just didn't say "this equals 3" since it was 3 which I stated I was proving. I assume that the reader doesn't forget what I stated 6 equations ago.

Thanks for pointing it out though. I think that just might make it clearer.

I was wondering if any of you folks have heard the terms Active Gravitational Mass and Passive Gravitational Mass? Theoretically they are proportional to each other. You can choose constants of proportionality such that they're differerent, However, regardlesso how you chose those constants the densities of these quantities have the same value.

A longtime ago I wrote a web page addressing this matter. I placed it here

http://home.comcast....e_grav_mass.htm

The density is seen in Eq. (3). Let rho = density of active gravitational density. As you can see from

Eq. (3) rho_{active g-mass} = u₀ + 3p. I don't derive it on that page (but will create a new page soon) but the passive gravitational mass density equals rho_{passive g-mass} = u₀ + p. It is the case tht passive g-mass = active g-mass. I think that's equal by definition. Not sure. I forgot.

The point I'm making is that even though you can choose constants so that they're equal, there is a very good reason to define two equantites since they have different densities in general.

Consider now a particle moving in a static gravitational field. The 4-momentum of the particle is P = (mc, p) where m is the passive gravitational mass as well as the inertial mass. Then for a particls at rest in the field m² = P². The energy if the particle is P₀ where the mass of the particle is The energy if the particle is P⁰.

Note: m(0) is the rest mass butit is not equal to the particles proper mass. I.e. m(0) != |P|/c. In this thread I'm hoping to make it clear that one has to be careful with what they call mass in SR/GR. It's not as easy as it appears.

Pete

ps - When I create a new page on the subject I'll post a URL to it if there is a desire for somonme to read it.

Which is, by definition, non-Newtonian, and is what I meant.....

I'm not sure what you mean when you said Which is, by definition, non-Newtonian, and is what I meant. Can you explain it for me? What is it that is non-Newtonian?

In GR it isn't a force, ...

I disagree. Gravity is an inertial force and since Einstein built GR around the idea that the gravitational force was an inertial force and that inertial forces are real and GR was Einstein's theory then in GR gravity is a force.

Well known experts in GR hold this same thing even today. For example. See Was Einstein Right? Putting General Relativity to the Test. Clifford M. Will - Updated and Newly Revised. See page 94

... has its counterpart here in a velocity dependant modification of the gravitational force.

Here, Will is speaking about the velocity dependance of all forces in SR and how it has its counterpart in GR.

If gravity is a fictitious (pseudo) force, cp. centrifugal & Coriolis forces;

To answer your question one needs to know what an inertial force is.

Inertial force - A force that must be added to Newton's equations of motion when using a non-inertial frame of reference so that Newton's laws will still be valid. See the link some comments and references on this subject.

The centrifugal force and the Coriolis force are psuedo-force and fictitious force. These terms are quite misleading so it is suggested that they not be used. For details please see http://home.comcast.net/~peter.m.brown/gr/inertial_force.htm

According to Einstein, the gravitational force is as real as the electric force and as such it is seen to be real.

Albert Einstein - That the relation of gravity to inertia was the motivation for general relativity is expressed in an article Einstein wrote which appeared in the February 17, 1921 issue of Nature

 

Can gravitation and inertia be identical? This question leads directly to the General Theory of Relativity. Is it not possible for me to regard the earth as free from rotation, if I conceive of the centrifugal force, which acts on all bodies at rest relatively to the earth, as being a "real" gravitational field of gravitation, or part of such a field? If this idea can be carried out, then we shall have proved in very truth the identity of gravitation and inertia. For the same property which is regarded as inertia from the point of view of a system not taking part of the rotation can be interpreted as gravitation when considered with respect to a system that shares this rotation. According to Newton, this interpretation is impossible, because in Newton's theory there is no "real" field of the "Coriolis-field" type. But perhaps Newton's law of field could be replaced by another that fits in with the field which holds with respect to a "rotating" system of co-ordinates? My conviction of the identity of inertial and gravitational mass aroused within me the feeling of absolute confidence in the correctness of this interpretation.

A.P. French - Inertial force is defined as the force on a body that results solely from observing the motion of the body from a non-inertial frame of reference. This in addressed in Newtonian Mechanics, A.P. French, The M.I.T. Introductory Physics Series, W.W. Norton Pub. , (1971) , page 499. After describing the inertial force as seen from an accelerating frame of reference French writes

 

From the standpoint of an observer in the accelerating frame, the inertial force is actually present. If one took steps to keep an object "at rest" in S', by tying it down with springs, these springs would be observed to elongate or contract in such a way as to provide a counteracting force to balance the inertial force. To describe such force as "fictitious" is therefore somewhat misleading. One would like to have some convenient label that distinguishes inertial forces from forces that arise from true physical interactions, and the term "psuedo-force" is often used. Even this, however, does not do justice to such forces experienced by someone who is actually in the accelerating frame of reference. Probably the original, strictly technical name, "inertial force," which is free of any questionable overtones, remains the best description.

The resistance of a gravitational field depends on the frame of reference that one is using. Just as frames of reference can change, so too is the existence of the gravitons. This strange notion is found in the relative existence of photons in a falling charged particle.

At this point I'd like to caution you regarding the presence of a gravitational field and curved spacetime. The presence of a gravitational field is determined by the existence of a non-inertial frame of reference, not tidal gradients. Therefore the presence of as a gravitational field is determined by the the presence is the affine components in a spatial Cartiansian coordinate system.

That is just the American Physicist Society definition of science, which misses whole scientific disciplines as synthetic chemistry. The definition of science given before was developed to improve that.

That is simply incorrect. Just because the name of the organization which published the article has the word physics in it can in no way be used as a valid excuse of it being the definition of science. The example you gave merely shows that synthetic chemistry, as you've used it, means that synthetic chemistry is not a valid branch of science.

That is just the American Physicist Society definition of science, which misses whole scientific disciplines as synthetic chemistry. The definition of science given before was developed to improve that.

Please illustrate your assertion with an example. Thanks.

Pete

No, massless particles having momentum can be derived in special relativity.

 

E²=(mc²)²+p²c²

 

Please post such a derivation. Typically such derivations are really identities.

I've never seen the inverse square law applied to relativistic bodies before, since it's generally only accurate at low velocities. This is interesting though.

I'm delighted to see that you found it interesting. What are you refering to when you said above since it's generally accurate at low velocities.

This can be found in the general relatiivity way back. Since it was already done by the time the 60's had come along, nobody had thought it worth while to do since it was alredy well known. Then things started to change and the literature became more and more different than the past and so the body of the material had changed quite a lot. For example; things like the gravitational field of a directed beam of light came into the literature. See http://home.comcast..../grav_light.htm

I had seen these things becomming less and less popular so I put them on my own website. But on my site I decided to make things like a long moving rod. See http://home.comcast...._moving_rod.htm

They're very interesting to see and play with. Take the metrics from the web site and copy them on a sheet of paper. The slow the particles down and make the field very very week. You should then see that the results become the Newtonian limit. There's the beginning for one for a sheet of matter.

On this page you can see the active gravitational mass in the equations. See http://home.comcast....e_grav_mass.htm If you look carefully you can see the proper mass density and the pressure and how it fits into the equations. If you were to compare this equation with the ones found in general relatvity texts that is what you'll see. Perhaps you might even see it in Schutz's book Gravity From the Ground Up. Some of it is online at Gravity From the Ground Up at http://www.gravityfromthegroundup.org/. You'll see those equations in his GR book as well as Peebles book on Cosmology too. If you take a look at inertial mass you'll see only one term for gravity whereas for a black hole there are there terms. Want to guess why? I've see general relativity experts miss that. This is a very good reason for knowing the hows and whys of relativistic mass. You have to know why the answer you get is the way it is. If you see the term "2" instead of, say, "3" you might miss the correct answer. You see, it's not enough to merely get the answer to a problem, but you have to know whether the answer you got is the correct answer.

Pete

Santalum, my answer to your question concerning gravity and the indentation in a rubber sheet is as follows. Yes it is only an analogy that directs human thought away from having to imagine orbits to that of a visual experience.

That's not quite correct. The embedding diagram for a Schwarzschild geometry can be found in the book Exploring Black Holes by Taylor and Wheeler which is at http://www.eftaylor.com/pub/chapter2.pdf (Moderator - Don't worry. These sections are available to be placed on line by the authors). Download and turn to page 2-26. Look at Figures 6 and 7. Read what is stated in the diagrams for for r, dr, d(sigma). If a collection of rulers was layed down around the Black Hole then you would see a different set of distances than would be there had not the black hole been present.

Pete

The following is the definition of science from the America Associatio of Physics Teachers. This is not copywritten material.

http://home.comcast.net/~peter.m.brown/ref/what_is_science.pdf

I prefer this definition to any other I hae been able to find. I hope that I don't get in trouble for pasting this here.

Pete