Potential Energy Explained: Does It Contradict Conservation of Mass/Energy?

[learningphysics]

The integral gives you a change in potential, not the absolute potential at a point. The change in potential from infinity to infinity is zero.

That's true. But we can define an absolute potential by fixing the lower limit at a specific value.

 

[learningphysics]

So how come that Gokul got a nonzero value of potential at R--> infinity?

Gokul never used absolute potential. He calculated the difference between the potential at the center and the potential at infinity.

So he never calculated potential at R--> infinity.

 

[Cyrus]

Ok. So if your saying that he calculated the difference in potential, don't we automatically know that the phi_infinity term, the second term being subtracted is zero. So the only term we have to consider is the potential at the center of the earth. And I thought that its value of potential energy is,: -GmM/R, no matter where you are inside the earth, so the associated potential would be -GM/R, factoring out the mass. But he got a different anwser.

 

[learningphysics]

Ok. So if your saying that he calculated the difference in potential, don't we automatically know that the phi_infinity term, the second term being subtracted is zero.

Right. It doesn't have to be zero. But we can certainly define it as being zero. So if we do define it as zero, the potential at the center is the expression on the right hand side of the equal sign.

So the only term we have to consider is the potential at the center of the earth. And I thought that its value of potential energy is,: -GmM/R, no matter where you are inside the earth, so the associated potential would be -GM/R, factoring out the mass. But he got a different anwser.

When you are inside the earth, you have to be careful. The gravitational field at 0<r<R is only due to the mass inside the radius r. You have to calculate the mass inside a sphere of radius r... and see what the gravitational field due to that mass is. So the mass that's creating the gravitational field at 0<r<R is a function of r... it is the density of the Earth times the volume.
$$\rho=\frac{M}{\frac{4}{3}\pi R^3}$$.
$$m=\frac{4}{3}\pi r^3 \rho$$

So mass comes out to: $$\frac{Mr^3}{R^3}$$

Gravitational field comes out to: $$-\frac{GMr}{R^3}$$

Integrate this from r=R to r=0, and we get: $$-\frac{GM}{2R}$$

Add this to the integral of the gravitational field from r=infinity to r=R, and we get: $$-\frac{GM}{2R} + -\frac{GM}{R}$$

so we get Gokul's answer: $$-\frac{3GM}{2R}$$

So if we put a mass m at the center of the earth... and if we take the potential at infinity=0, then the gravitational potential energy would be:
$$-\frac{3GMm}{2R}$$

 

[Cyrus]

Thanks for your help. I must say, that certainly clears things up. :-)

 

[learningphysics]

Thanks for your help. I must say, that certainly clears things up. :-)

Cool. Glad to help.

Sleepy time now! :zzz:

 

[michael879]

No, Andrew is right. GPE is zero at infinity and goes negative as you approach the Earth. That is because it is expressed as
$$GPE =- \frac{GMm}{r}$$
( Note: G = the gravitational constant and is not g which is the acceleration due to gravity.)

oops, I forgot the -

 

[Twukwuw]

the conclusion...

After reading all the posts I now draw some conclusions:
:yuck:
1)When the total energy of an object is changed, its RELATIVISTIC mass must also change.

This is because E = MC^2 for an object, where E is the TOTAL relativistic energy of the object (as observed by an observer), M is the relativistic mass (M = rest mass x gamma), C is the light speed.
(Note that I emphasize that all the terms must be relativistic, because object A may have different amount of energy in reference frames of object B and object C, due to the different relative velocities.)
Since the relativistic mass is a function of m (rest mass) and v (relative velocity), it turns out that the energy level/total energy of an object is also a function of m and v, i.e.
E = f (m,v). Hence, when M is increased, m MAY BE INCREASED but also MAY BE DECREASED, provided there is a greater increase in v when M increases but m decreases!

2)When energy is created, there must be an amount of RELATIVISTIC mass being destroyed, or lost.

For example, during chemical reactions the energy released is related to the lost of RELATIVISTIC mass by ΔE = ΔMC^2. The lost of M can be interpreted this way:
Case 1: The velocity of each chemical particle decreases( averagely ), and M therefore decreses because the relativistic mass of each particle has decreased.
Case 2: The rest masses of chemical particles decrease SOMEHOW!
Case 3: Both the case 1 and case 2 are involved!
With this in mind, we realize that when one does work (which in fact the energy comes from complicated chemical reactions inside his bodies), his RELATIVISTIC mass (with respect to any reference frame) must decrease!
Other than that, when we gain heat energy from a heat reservoir, the relativistic mass of the heat reservoir will also decrease! Why? Because when we extract energy from the heat reservoir, its temperature ( a measure of the average kinetic energy, or average velocity of each particle.) must decrease, hence the relativistic mass must decrease! (What I want to point out here is, when the temperature of an object is changed, its relativistic mass will be also changed.)

3)The potential energy is not a “real” one, it is just created for mathematic convenience.

When we say an object has an extra energy potential energy, we are saying that IN THE FUTURE it will acquire an amount of energy which is the same as the potential difference. And, the energy comes from other resource; here it is the MASS who set up its gravity field, but NOT the gravity field (or the curvature of space-time)!
To give a clearer picture, here I give an “analogy…”
Suppose there is a box in a zero-gravity-field region in the outer space. Inside the box there are 2 identical stone and MOST importantly a person.
It is a rectangular box. Now, stone A is at a higher level, while stone B and the person are at the “bottom” of the box. Both the stones are STATIONARY. First we ask, with respect to the person, are the energies of the 2 stones different? Not! Why? Because they have the SAME REST MASS and the SAME RELATIVE VELOCITY (quickly linked to the situation in the Earth in which 2 identical stones are at different height levels!)
Now, the person say: I will do a work of magnitude ΔW on the stone B such that it will reach stone A and at the same time having a velocity V. When we are informed about the person’s decision, we now say:
BESIDE ITS REST ENERGY, the stone B now also have POTENTIAL ENERGY of ΔW!
Why? Because the person PROMISES that he will give stone B an extra energy in the FUTURE!
So, that is all! That is what we mean by potential energy! It is nothing but merely a term INFORMING us that in the FUTURE an extra energy will be given to stone B!
Now, let’s again have a question, are the masses of the 2 stones different because stone B have a potential energy? NO! They are the same! (Again, quickly linked to the situation in which 2 identical stones are at different height levels, do they have different relativistic masses? You know it! )

Well, these are my opinions and viewpoints. However, I understand that I may be mistaken. Please feel free to write back to me if you find me wrong!
Thank you for your reading!

 

[Janus]

.

I'll just address a couple of quick points here.

This is because E = MC^2 for an object, where E is the TOTAL relativistic energy of the object (as observed by an observer), M is the relativistic mass (M = rest mass x gamma), C is the light speed.
(Note that I emphasize that all the terms must be relativistic, because object A may have different amount of energy in reference frames of object B and object C, due to the different relative velocities.)

No, in the formula
$$E = mc^2$$

m represents the rest mass of the object and E the energy equivalence of that rest mass.

The equation that gives the total energy of the object (and the equation from which E=mc² is derived) is

$$E_t =\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$

The equation that would give the kinetic energy for the object would be

$$KE =mc^2 \left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1 \right)$$

2)When energy is created, there must be an amount of RELATIVISTIC mass being destroyed, or lost.

'Relativistic mass' and 'energy' are just different terms for the same thing. In fact, the term relativistic mass has fallen into relative disuse among scientists and they use the term energy almost exclusively instead.

 

[learningphysics]

Now, let’s again have a question, are the masses of the 2 stones different because stone B have a potential energy? NO! They are the same! (Again, quickly linked to the situation in which 2 identical stones are at different height levels, do they have different relativistic masses? You know it! )

Well, these are my opinions and viewpoints. However, I understand that I may be mistaken. Please feel free to write back to me if you find me wrong!
Thank you for your reading!

The potential energy is in the person's body. When he does work on the stone, he loses rest mass. The person's body will weigh less than before he did work on the stone.

Note that for a person at rest... relativistic mass and rest mass are the same.

Potential energy in the body is equivalent to rest energy.

 

[catch.yossarian]

What if you look at it from another way...

Saying that something accelerating, and thus speeding it up, towards Earth is increasing its [Kinetic] energy (and thus losing Potential Energy) , and thus something must lose mass to compensate.

Isn't that like saying: "okay, since I'm adding a ton of mass to this object, it will suddenly speed up and fly away from here." ? This doesn't happen, though...

Or maybe I'm not even interpreting this thread. :\

 

[El Hombre Invisible]

Say you had two identical blocks of radioactive uranium and you put one at the bottom of the ocean and one at the top of mount everest. Now as rest mass is converted into energy, would you expect the one with the greater rest mass - the one at the top of the mountain - to yield more energy than the one at the bottom of the ocean?

 

[whozum]

Yaeh I've got to agree with yossarian, I really don't see where this thread is going. In the beginning we were talking about the translation of potential energy to kinetic energy, and now were bringing about relativity and rest masses, but I don't really see their connection to the initial question.

A body's energy due tohaving mass is independent of its energy due to position in a field..

 

[learningphysics]

Yaeh I've got to agree with yossarian, I really don't see where this thread is going. In the beginning we were talking about the translation of potential energy to kinetic energy, and now were bringing about relativity and rest masses, but I don't really see their connection to the initial question.

A body's energy due tohaving mass is independent of its energy due to position in a field..

If 1 kg of ice at 0C melts into water at 0C... the water gains mass.

The molecules in the ice or the fields between the molecules (I'm not sure which) weigh more in the water state. But total rest mass increases and this increase is due to the increase in potential energy.

 

[learningphysics]

Say you had two identical blocks of radioactive uranium and you put one at the bottom of the ocean and one at the top of mount everest. Now as rest mass is converted into energy, would you expect the one with the greater rest mass - the one at the top of the mountain - to yield more energy than the one at the bottom of the ocean?

Where gravity is concerned I'm not sure...I don't know anything about general relativity. I'm not sure whether the one at the top of the mountain has greater rest mass.

If we were in an analogous situation with an electrostatic field rather than a gravitational field... I'd say yes, the one at the further distance yields more energy.

 

[learningphysics]

Perhaps this link will be helpful to explain the connection between rest mass and potential energy:

http://musr.physics.ubc.ca/~jess/p200/emc2/node6.html

Note the example with regards to the ship leaving the earth.

 

[pmb_phy]

Perhaps this link will be helpful to explain the connection between rest mass and potential energy:

http://musr.physics.ubc.ca/~jess/p200/emc2/node6.html

Note the example with regards to the ship leaving the earth.

Here is an example from Nuclear Physics that I worked out as a specific example

http://www.geocities.com/physics_world/sr/nuclear_energy.htm

The mass of a particle depends on the position of the particle in a gravitational field (e.g. is a function of the gravitational potential). This works for gravity but does not work for any other field. Here by "mass" I mean m = p/v , not proper mass (aka "rest mass"). For details you can find the derivation in Einstein's text The Meaning of Relativity. Page 100 as I recall. This is Mach's Principle as Einstein saw in this text.

Pete

 

[pmb_phy]

No, in the formula
$$E = mc^2$$

m represents the rest mass of the object and E the energy equivalence of that rest mass.

That depends on how one defines mass and the symbol. Twukwuw quite clearly and explicitly defined E/c^2 as "relativistic mass." When the object is a particle or an isolated body then that is quite true.

'Relativistic mass' and 'energy' are just different terms for the same thing.

That is quite incorrect. They most definitely are not the same thing in different units. In fact they are not even proportional in general. If you want to read more on this see Rindler's intro sr text.

In fact, the term relativistic mass has fallen into relative disuse among scientists and they use the term energy almost exclusively instead.

That's not quite accurate either. For a counter example please see

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Seems to me that you're thinking mostly of particle physicists and not physicists in general. A search of the internet and a look at some modern relativity texts confirms this. Even particle physicists are being lazy when they do this. Its simply easier to do. They do the same thing with proper time. For example; if you look up the "lifetime" of a neutron in the particle physics literature you'll see that its about 15 minutes. However this is understood to be the "proper lifetime." But constanly using the adverb "proper" is a pain in the butt when everyone understands what it is. Since particle physicists study only the intrinsic properties of particlces its also a pain in the butt. Particle physicists don't study relativity in all its glory. In general the two cannot be interchanged. Especially for extended non-isolated bodies. In such cases the relation E/c^2 = p/v its incorrect and therefore E is not proportional to rel-mass and therefore it cannot be said that they're the "same thing."

I mentioned all this to Dr. David Morin at Harvard. He's the prof there who wrote that great set of lecture notes. He seems to agree on this and he will be going over it during the summer.

In the past I held that it was simply a matter of definition. However I've changed my mind. Long story and I want a cup of tea right at the moment!

Pete

 

[El Hombre Invisible]

learningphysics, I now see where our paths were crossed. I believe the original question of whether potential energy effected rest mass was about one part of the system within which the interactions were occurring. For instance, when a charge A moves away from an opposite charge B, the rest mass of A remains unchanged, likewise for B. However the mass of the system as a whole, A+B, increases due to an increase in potential energy. This, it seems, is what your link refers to. The way I understand it, the rest mass of a system can be deemed to be the relativistic masses of all of its components in a frame of reference such that the system as a whole is at rest. As relativistic momentum includes those of force-carrying particles, the potential energy of all interactions within the system are included in the rest mass of the system as a whole.

 

[El Hombre Invisible]

And then I read to the bottom: "As a result, we might expect the rest mass of a spaceship to be slightly larger after it leaves the Earth than it was on Earth, simply because it has left the "gravity well'' of the Earth. This is the case! However, the mass change is imperceptibly small in this case. " D'oh! Now I go back to confusion again. Woe is me!

 

[Cyrus]

If 1 kg of ice at 0C melts into water at 0C... the water gains mass.

The molecules in the ice or the fields between the molecules (I'm not sure which) weigh more in the water state. But total rest mass increases and this increase is due to the increase in potential energy.

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

 

[pmb_phy]

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

learningphysics is correct. The relation you posted, E=K+mc^2, is the inertial energy of a single point particle which is moving. He is not speaking of such a particle. He is speaking of water. One can think of the water as balls connected by springs. The balls are the H and O atoms and the springs represent the mutual electric force between the balls. All the balls are moving and the springs are compressing and expanding. There is a total inertial energy for each ball and a potential energy for the springs (the potential energy well is not symmetric as I recall). An increase in energy will then cause an increast in the total energy (kinetic energy of all balls + potential energy of all springs).

Simply put - The ice must absorb energy for it to melt. Any change in energy of the ice must result in an increase in the mass, and hence weight, of the H₂O.

Pete

 

[pmb_phy]

For instance, when a charge A moves away from an opposite charge B, the rest mass of A remains unchanged, likewise for B. However the mass of the system as a whole, A+B, increases due to an increase in potential energy.

The mass of the system as a whole remains unchanged. The magnitude of the potential energy decreases with seperation. Whether potential is negative or positive and whether the totak kinetic increases or decreases depends on the signs of the charges. The sum of kinetic and potential energy remains the same. This, of course, neglects the radiation due to acceleration of the charges.

What has not yet been addressed is the mechanism of the increase due to potential energy. Nor has the mechanism due to the increase in weight been addressed. I wrote a paper on all of this and will post it on my web page tommorow. It will explain these mechanisms.

Pete

 

[learningphysics]

learningphysics is correct. The relation you posted, E=K+mc^2, is the inertial energy of a single point particle which is moving. He is not speaking of such a particle. He is speaking of water. One can think of the water as balls connected by springs. The balls are the H and O atoms and the springs represent the mutual electric force between the balls. All the balls are moving and the springs are compressing and expanding. There is a total inertial energy for each ball and a potential energy for the springs (the potential energy well is not symmetric as I recall). An increase in energy will then cause an increast in the total energy (kinetic energy of all balls + potential energy of all springs).

Simply put - The ice must absorb energy for it to melt. Any change in energy of the ice must result in an increase in the mass, and hence weight, of the H₂O.

Pete

But rest mass only increases with an increase in potential energy right (not KE)?

 

[learningphysics]

The mass of the system as a whole remains unchanged.

But this is relativistic mass that you're referring to right? Doesn't rest mass change with potential energy?

 

[learningphysics]

I don't think that's true learningphysics. E=K+mc^2. Melting the ice increases the internal kinetic energy of the water molecules. So K goes up to compensate for the increased energy and not the mass. no?

Both the water and the ice are at 0C... so the kinetic energy is the same. Water at 0C has more potential energy than ice at 0C. When you heat up ice the temperature goes up until you get to the melting point 0C... then all the energy goes into potential energy which leads to the melting of the ice.

Increased potential energy, I believe means increased rest mass.

 

[learningphysics]

And then I read to the bottom: "As a result, we might expect the rest mass of a spaceship to be slightly larger after it leaves the Earth than it was on Earth, simply because it has left the "gravity well'' of the Earth. This is the case! However, the mass change is imperceptibly small in this case. " D'oh! Now I go back to confusion again. Woe is me!

Yes... I really don't know where exactly the increased rest mass resides.

 

[pmb_phy]

But rest mass only increases with an increase in potential energy right (not KE)?

No.

But this is relativistic mass that you're referring to right?

It makes no difference.

Doesn't rest mass change with potential energy?

No.

The above assumes that "rest mass" referres to the mass of a closed system as measured in the zero momentum frame.

The following may help (what you call rest mass others call "invariant mass". The usage is not really consistent in the literature.)
http://www.geocities.com/physics_world/sr/invariant_mass.htm

This is a perfect worked example of what this thread is about
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Pete

 

[El Hombre Invisible]

The mass of the system as a whole remains unchanged. The magnitude of the potential energy decreases with seperation. Whether potential is negative or positive and whether the totak kinetic increases or decreases depends on the signs of the charges. The sum of kinetic and potential energy remains the same. This, of course, neglects the radiation due to acceleration of the charges.

What has not yet been addressed is the mechanism of the increase due to potential energy. Nor has the mechanism due to the increase in weight been addressed. I wrote a paper on all of this and will post it on my web page tommorow. It will explain these mechanisms.

Pete

Hi Pete. First off, if the system is closed then the relativistic mass of the radiation is part of that system and so would be counted towards the relativistic mass of the system as a whole, so it should not be neglected.

In my example, I specified two opposite charges - let's say a proton and an electron. As the electron moves away from the proton, the strength of the field, or the intensity of the photons interacting with each, should increase and the kinetic energy of the moving charge will decrease. This will lead to a decrease in the relativistic mass of the electron compensated for by the increase in the energy of the field, no? If this equal potential energy were stored within the electron itself, rather than in the field, there would be no observable decrease in its relativistic mass, as ΔEpot - ΔEtrans would be 0, and so E would be constant. If you then open this up to all interactions the electron takes part in, then in any frame of reference the relativistic mass of the electron would remain constant, as any change in position would be compensated for my a corresponding change of potential energy, and the relativistic mass would no longer be effected by velocity, which would seem inconsistent with SR. It would also appear to defy the laws of conservation of energy, as if the relativistic mass of both the proton and the electron remained constant during separation, and the intensity of the photons in between increased, where does this increase of energy come from?

 

[pmb_phy]

Hi Pete. First off, if the system is closed then the relativistic mass of the radiation is part of that system and so would be counted towards the relativistic mass of the system as a whole, so it should not be neglected.

I neglected if for simplicity assuming that it is neglegible compared to the kinetic and potential energy. Its simple to add it in but it was easier to speak of. Not to mention it seemed what you were doing since you also never mentioned the radiation in your example and it was your example I was addressing.

In my example, I specified two opposite charges - let's say a proton and an electron. As the electron moves away from the proton, the strength of the field, or the intensity of the photons interacting with each, should increase and the kinetic energy of the moving charge will decrease. This will lead to a decrease in the relativistic mass of the electron compensated for by the increase in the energy of the field, no?

As the distance between the charges increases to infinity the total potential energy of the system increases towards zero and the kinetic energy decreases toward zero. The total energy of the system remains constant. Therefore the total mass of the system remains constant. It'll be easier for you to see all this by the principle of the conservation of energy. Since E = mc^2 then since E = constant then so does m. However when you actually calculate this it is the potential energy which comes into the energy calculation.

If this equal potential energy were stored within the electron itself, ...

No. That is not true. The mass is in the system itself and is not "stored" in one of the particles.

It would also appear to defy the laws of conservation of energy, as if the relativistic mass of both the proton and the electron remained constant during separation, and the intensity of the photons in between increased, where does this increase of energy come from?

You've added in a new complication when you started to take into account the radiation. This is a very tricky question and I believe that the answer is related to the self force acting on each charge so you no longer have just the Coulomb force acting (and the associated potential energy). I'm only moderately familiar with the physics so I'll be quite on this point for now (stuff this complicated I forget a week after I figure it out/learn it! ). I believe I can dig that up somewhere though. If this is related to the problem called "mass renormalization" then that is really really really tricky and requires some really advanced stuff to give an answer to. That seems to go far beyond what you're looking for since you don't care about that - you care about potential energy and its relationship to mass. Recall Einstein's first derivation - A body can absorb radiation and when it does the mass increases. What happens inside the body is the electromagnetic energy is changed to internal potential energy. This then is the energy associated with the increase in mass.

If you consider two like charges at rest - find the mass - then move the charges closer together and then let them be at rest once again then the mass of that system increases. The mechanism which causes the increase in mass can seen by weighing the system. Each charge exerts a force on the other charge and the force has a negative component, i.e. in the direction of the g-field. This added force gives an added weight.

Here's a tricky one for you - Consider a point charge. What is the mass? You'll have to address the mass of the field and the intrinsic mass of the particle itself. Note that the mass of the field is infinite.

If you're truly interested in find and answer to your questions then try to imagine an experiment which will measure the mass you're speaking of.

Pete

Note; Speaking about the field itself as if it had a mass is very tricky and will mess you up big time. The mass associated with an EM field of a chared particle will not transform in the same way as the mass of a particle.

 

[learningphysics]

No.
It makes no difference.
No.

The above assumes that "rest mass" referres to the mass of a closed system as measured in the zero momentum frame.

The following may help (what you call rest mass others call "invariant mass". The usage is not really consistent in the literature.)
http://www.geocities.com/physics_world/sr/invariant_mass.htm

This is a perfect worked example of what this thread is about
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Pete

Thanks for your help Pete. Your website is great.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases... whether or not the energy went into increasing the kinetic energy of the constituent particles... or the potential energy of the constituent particles.

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

 

[pervect]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously). Steve Carlip, for instance, has a paper on the issue of gravitational mass of a system (which is trickier than it first appears BTW), but it definitely takes more than simple SR to understand the paper. The abstract, though, at least clearly states that kinetic energy does contribute to the gravitational mass of a system of particles - so you may have to just take this on faith a bit, unless and until you get the needed background.

Kinetic Energy
and the Equivalence Principle
S. Carlip∗
Department of Physics
University of California
Davis, CA 95616
USA
Abstract

According to the general theory of relativity, kinetic energy contributes
to gravitational mass. Surprisingly, the observational evidence for this
prediction does not seem to be discussed in the literature. I reanalyze
existing experimental data to test the equivalence principle for the
kinetic energy of atomic electrons, and show that fairly strong limits
on possible violations can be obtained. I discuss the relationship
of this result to the occasional claim that “light falls with twice the
acceleration of ordinary matter.”

 

[pmb_phy]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously).

"really" understand? If one understands it without GR does that mean neccesarily that he "really" doesn't understand it? The first derivation I ever did to determine whether a moving body weighed more I did before I learned GR to any great extent. All one requires is the equivalence principle. From that one can extrapolate to a more complex object such as a 2-d gas in a 2-d box. If one recalls Einstein's 1905 SR paper then in it Einstein states that to measure the transverse mass of a particle one can use a balance. Since the transverse mass = relativistic mass one obtains the weight of the particle. Its always best to try to figure these things out before one puts pen to paper. I've rarely made any progress in anything I've ever done by starting with the math.

Pete

 

[pmb_phy]

Thanks for your help Pete. Your website is great.

Thank you.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

Join the club.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

That is correct.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Sure. As long as you're talking about non-interacting particles. You don't even need to employ energy conservation. This fact can be derived without ever employing the principle of the conservation of energy. Using the theory of the conservation of mass makes it trivial.

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases...

Brigtht boy!

For a derivation of what you just stated see
http://www.geocities.com/physics_world/sr/mass_energy_equiv.htm

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

See -
http://www.geocities.com/physics_world/sr/weight_moving_sr.htm
http://www.geocities.com/physics_world/gr/weight_move.htm

Pete

 

[El Hombre Invisible]

Pete - I now realize that it made no sense to address my post to you. We seem to be in complete agreement, bar the point about the rest mass of the system where I temporarily went insane - you are of course 100% correct.