Is Relativistic Mass Still Relevant in Modern Physics Discussions?

[Aer]

Yes a photon has energy.

Does it have mass?

 

[Aer]

Physicists generally define "inertial mass" in terms of resistance to acceleration in the object's own rest frame, and you can't do this for a photon, although you can do it for a compound system which contains a photon.

OK - find all the inertial masses of the object's by themselves in their own rest frames.

 

[JesseM]

Just like I thought, you'd come up with another BS answer.

So do you disagree with me that the debate over "relativistic mass" is an aesthetic one rather than a matter of different predictions? Do you think someone using relativistic mass will make a different physical prediction than someone who doesn't?

You can't even keep your arguments consistent! Pick a theory and stick with it. Either all energy contributes to an objects mass or it does not (and I am referring to the macroscropic world here).

Did you read my post #146 from 12:39 PM? It depends on whether you use "mass" to mean rest mass or inertial mass, my arguments are consistent once you understand which one I'm talking about in which cases.

If you claim that quantum physics is the same regarding mass and energy as is on the macroscopic world, then the mass of an object in quanutm physics would be the total energy / c^2. I don't dispute the latter, it is the former that I dispute. That is - on the macroscopic level, other forms of energy exist other than mass energy.

Do you dispute the fact that our current theories of physics make a definite prediction about this, and that they say that the resistance to acceleration of a compound object is proportional to its total rest energy?

 

[Aer]

So do you disagree with me that the debate over "relativistic mass" is an aesthetic one rather than a matter of different predictions? Do you think someone using relativistic mass will make a different physical prediction than someone who doesn't?

What prediction would you like to make?

 

[JesseM]

OK - find all the inertial masses of the object's by themselves in their own rest frames.

You can't do this for a photon, but you can do this for any object moving slower than light. What's your point? The inertial mass of a compound object will not be the sum of the inertial masses of all the objects that make it up, according to relativity (assuming, again that you use the words 'inertial mass' to refer only to resistance to acceleration in the object's rest frame--if you allow the words 'inertial mass' to refer to resistance to acceleration in other frames, then the inertial mass of a compound object in its own rest frame is the sum of the inertial masses of all its parts in that frame, assuming there is no potential energy between the parts).

 

[learningphysics]

Does it have mass?

It has zero rest mass. It has relativistic and inertial mass = h\nu/c^2

 

[JesseM]

So do you disagree with me that the debate over "relativistic mass" is an aesthetic one rather than a matter of different predictions? Do you think someone using relativistic mass will make a different physical prediction than someone who doesn't?

What prediction would you like to make?

Personally I would expect the theory of relativity is correct in its prediction that the resistance to acceleration of a compound object is proportional to its total energy. Again, do you dispute that this is what the theory of relativity would predict? Please answer this question yes or no.

 

[Aer]

It has zero rest mass. It has relativistic and inertial mass = h\nu/c^2

Does gravity act on this inertial mass?

 

[learningphysics]

Does gravity act on this inertial mass?

Yes. Gravity bends light.

 

[Aer]

Personally I would expect the theory of relativity is correct in its prediction that the resistance to acceleration of a compound object is proportional to its total energy.

Let's talk about a single object first. What do you expect relativity to predict about a single object?

 

[Aer]

Yes. Gravity bends light.

The curvature of space bends light.

 

[JesseM]

Let's talk about a single object first. What do you expect relativity to predict about a single object?

A single particle? It would predict that its resistance to acceleration in its own rest frame (assuming it's a sublight particle) is proportional to its rest mass. Now will you answer my question about what relativity predicts for a compound object?

 

[Aer]

A single particle? It would predict that its resistance to acceleration in its own rest frame (assuming it's a sublight particle) is proportional to its rest mass.

If an object is moving relative to me at .9c, what would I predict it's mass to be?

Now will you answer my question about what relativity predicts for a compound object?

One step at a time. Look above.

 

[JesseM]

If an object is moving relative to me at .9c, what would I predict it's mass to be?

What do you mean by "mass"--rest mass? Relativistic mass? Inertial mass in the object's own rest frame? Inertial mass in your own rest frame? (physicists who prefer to avoid using 'relativistic mass' will want to avoid using this last concept of inertial mass too)

 

[Aer]

What do you mean by "mass"--rest mass? Relativistic mass? Inertial mass in the object's own rest frame? Inertial mass in your own rest frame? (physicists who prefer to avoid using 'relativistic mass' will want to avoid using this last concept of inertial mass too)

I just want what SR predicts. A force should be able to move this object - so what is the mass?

 

[JesseM]

I just want what SR predicts.

You can't make a definite prediction unless you define your terms. Different physicists may use the term "mass" differently but they're still making use of the same theory of relativity--the choice of terminology is a matter of tradition and aesthetics, it's not a physical question. Hell, we could interchange the meaning of "mass" and "length" if we wanted, theories of physics don't demand that you use language in a particular way, although if different physicists use different terms they must know how to map one set of terms to another to make sure they are not disagreeing about any physical predictions.

A force should be able to move this object - so what is the mass?

The amoung of force needed to accelerate the object by a small amount will depend on what frame you're in.

 

[Aer]

The amoung of force needed to accelerate the object by a small amount will depend on what frame you're in.

Good, but there is only one force acting on the body in reality and the only proper frame to measure this in is the rest frame of the object which is subjected to the force, correct?

 

[Aer]

This discussion is going nowhere - let's assume that relativity does say that mass is dependent on the total energy content of a system. Then we have the problem of showing experimental proof to say that this is true.

Let's just assume that it is true as you state it. Now can we find any experiments?

 

[Aer]

I think pervect summed it up with his post pointing to: http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909014.pdf which states there is no experimental evidence to back up the assertion.

 

[learningphysics]

I think pervect summed it up with his post pointing to: http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909014.pdf which states there is no experimental evidence to back up the assertion.

No evidence for what? In his last sentence before the Acknowledgments he writes:

"We can thus tell our students with confidence that kinetic energy has weight, not just as a theoretical expectation, but as an experimental fact."

Do you agree or disagree with this?

 

[Aer]

No evidence for what? In his last sentence before the Acknowledgments he writes:

"We can thus tell our students with confidence that kinetic energy has weight, not just as a theoretical expectation, but as an experimental fact."

Do you agree or disagree with this?

Does that not imply that the kinematic energy must be measured relative to the rest frame of the gravitational potential? Otherwise, what is the meaning of kinetic energy? The object must have motion relative to something to have kinetic energy.

 

[εllipse]

I think pervect summed it up with his post pointing to: http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909014.pdf which states there is no experimental evidence to back up the assertion.

It also states that general relativity predicts kinetic energy is a part of gravitational mass. This is a relativity forum, and as such I think we should conclude that GR's predictions have the final say in this if such predictions have not been tested against experiment.

 

[Aer]

Perhaps the problem is we are trying to equate weight from gravity with mass.

From: http://www.conceivia.com/topics/not_quantum_physics.htm

At first, the assumptions made a certain amount of sense and were even believable. It wasn't until the assumption that kinetic energy has mass, that everything got all out of wack. I'm not saying that this assumption was incorrect, in fact I believe it was a valid assumption.

The mistake was incorporating this assumption into the formula for acceleration and decelleration. The mass gained from kinetic energy does not add to the kinetic energy. It adds to the gravitation of the moving object, but not to it's kinetic energy. The reason for this is that the mass of the object is relative to itself.

 

[JesseM]

Good, but there is only one force acting on the body in reality and the only proper frame to measure this in is the rest frame of the object which is subjected to the force, correct?

"Proper frame" by what criterion? You're free to analyze any situation in any frame you like according to relativity. But if you are asking about the force needed in the object's own rest frame, this will indeed be proportional to its rest mass for a single particle. Now can you answer my question about whether you agree or disagree that the theory of relativity predicts that for a compound object, the force needed to accelerate it a given amount in its rest frame will be proportional to its total rest energy?

 

[Aer]

"Proper frame" by what criterion? You're free to analyze any situation in any frame you like according to relativity. But if you are asking about the force needed in the object's own rest frame, this will indeed be proportional to its rest mass for a single particle.

Now wait - a certain amount of energy is known to be used to accelerate the object. Since the force changes depending on the frame, does that mean the energy changes?

 

[Aer]

the force needed to accelerate it a given amount in its rest frame will be proportional to its total rest energy?

"total rest energy" is ambiguous.

 

[pervect]

Now wait - a certain amount of energy is known to be used to accelerate the object. Since the force changes depending on the frame, does that mean the energy changes?

Definitely. This is true in Newtonian mechanics as well.

It takes much more energy to go from 10 m/s to 11 m/s than it does to go from 0 m/s to 1 m/s.

 

[Aer]

Definitely. This is true in Newtonian mechanics as well.

It takes much more energy to go from 10 m/s to 11 m/s than it does to go from 0 m/s to 1 m/s.

So when the acceleration is constant in the frame of our object, you are saying the energy required to accelerate it constantly increases in the frame of our object?

 

[Aer]

I wish I had the thread on hand, but the "relativistic mass" for a force from one frame to another was γ³m was it not? I think it was you who came up with this, though it may have been someone else - I forget. Anyway, in this thread, it was established that "relativisitic mass" was simply γm, where m is rest mass, was it not?

 

[jtbell]

What do you mean by "mass"--rest mass? Relativistic mass? Inertial mass in the object's own rest frame? Inertial mass in your own rest frame?

To elaborate: in classical physics, we can associate to any object a number m that has the following useful properties:

(a) we can use it to calculate the object's acceleration in response to a given force, \vec a = \vec F / m, regardless of how the object is moving to begin with, and regardless of the direction of the force. ("inertial mass")

(b) we can use it to predict the gravitational force that the object exerts on another object; also how the object responds to the gravitational influence of another object. ("gravitational mass")

(c) for any particular object, m is constant, and an intrinsic property of the object, so long as we're not adding pieces to the object or chipping pieces away from it. ("invariant mass")

(d) if we combine two objects together to form a single object or system, we can simply add m_1 + m_2 = m to get a number that plays the same role for the composite object.

In relativistic physics, no single number (or even a single formula that caculates a number as a function of speed) fills all of these roles. In particular, since nobody has mentioned it yet, I'd like to point out that (a) is especially problematical. Not only does an object's acceleration in reponse to a given force depend on how fast the object is moving to begin with, it also depends on the direction of the force relative to the object's direction of motion! The familiar formula for "relativistic mass" works only if the force is perpendicular to the direction of motion. If the force is parallel to the direction of motion, we have to use a different "relativistic mass". Some books call these "transverse mass" and "longitudinal mass". (And then of course, we have a "45-degree mass" and a "72-degree mass", etc. )

So, if you want to talk about the "mass" of an object in relativity, you have to specify, or at least have it already be understood from context, which of these properties you really want to deal with. You can't have them all.

 

[Aer]

So, if you want to talk about the "mass" of an object in relativity, you have to specify, or at least have it already be understood from context, which of these properties you really want to deal with. You can't have them all.

We were working with the assumption that the total energy (including kinetic) defined the mass of the system. And yes, it does appear problematic - but if you analyze things in the rest frame of the object, there is no problem.

 

[JesseM]

Now wait - a certain amount of energy is known to be used to accelerate the object. Since the force changes depending on the frame, does that mean the energy changes?

Presumably you can consider an infinitesimal acceleration from an infinitesimal input of energy, so you don't have to worry about this when defining resistance to acceleration in the object's own rest frame.

"total rest energy" is ambiguous.

Why? It's just the sum of the kinetic energy and rest masses of all the components with the potential energy between them as seen in the rest frame of the object as a whole.

 

[JesseM]

No evidence for what? In his last sentence before the Acknowledgments he writes:

"We can thus tell our students with confidence that kinetic energy has weight, not just as a theoretical expectation, but as an experimental fact."

Do you agree or disagree with this?

Does that not imply that the kinematic energy must be measured relative to the rest frame of the gravitational potential? Otherwise, what is the meaning of kinetic energy? The object must have motion relative to something to have kinetic energy.

Again, they're talking about the kinetic energy of parts of a compound object as seen in the compound object's rest frame. The introduction says:

The principle of equivalence—the exact equality of inertial and gravitational
mass—is a cornerstone of general relativity, and experimental tests of the universality
of free fall provide a large set of data that must be explained by any theory
of gravitation. But the implication that energy contributes to gravitational mass
can be rather counterintuitive. Students are often willing to accept the idea that
potential energy has weight—after all, potential energy is a rather mysterious
quantity to begin with—but many balk at the application to kinetic energy. Can
it really be true that a hot brick weighs more than a cold brick?

General relativity offers a definite answer to this question, but the matter is
ultimately one for experiment. Surprisingly, while observational evidence for the
equivalence principle has been discussed for a variety of potential energies, the
literature appears to contain no analysis of kinetic energy. The purpose of this
paper is to rectify this omission, by reanalyzing existing experimental data to look
for the “weight” of the kinetic energy of electrons in atoms. I will then try to
reconcile the results with the occasional (and not completely unreasonable) claim
that “objects traveling at the speed of light fall with twice the acceleration of
ordinary matter.”

Clearly the example of a hot brick vs. a cold brick involves a compound object whose parts can have greater or lesser kinetic energy in the object's rest frame, and likewise electrons in atoms are part of a compound object, so presumably he's talking about the gravitational mass (which by the equivalence principle is the same as inertial mass) of the atoms in their own rest frame, which can change as the kinetic energy of the electrons in this frame changes.

 

[EnumaElish]

Actually, isn't the m in E=mc^2 relativistic mass (if applied to other reference frames)? I thought the invariant mass version of the equation was E^2=p^2c^2+m^2c^4.

To my understanding, E=p^2c^2+m^2c^4=\gamma mc^2=m_r c^2; so yes, the m in E = mc² is relativistic mass. Aer would say that the quantity m_r should not be called "relativistic mass" because the second word of this definition starts with the 13^th letter of the contemporary English alphabet immediately followed by the 1^st letter of the said alphabet, and ends by repeating twice the 19^th letter of this symbolic collection. I guess an obvious alternative is to call it "shorthand for \gamma m."

 

[EnumaElish]

I wish I had the thread on hand, but the "relativistic mass" for a force from one frame to another was γ³m was it not? I think it was you who came up with this, though it may have been someone else - I forget.

You are referring to Doc Al's post under thread "Speed."

Anyway, in this thread, it was established that "relativisitic mass" was simply γm, where m is rest mass, was it not?

Yes, based on your own post and the following posts under the same thread.

 

[pervect]

I think pervect summed it up with his post pointing to: http://arxiv.org/PS_cache/gr-qc/pdf/9909/9909014.pdf which states there is no experimental evidence to back up the assertion.

The quoted reference says quite the opposite, actually.

 

[Aer]

The quoted reference says quite the opposite, actually.

I think the problem here is, we are referring to two very different things. Gravity effectively curves space which is why light bends in a gravitation field although there is no actual force on the photon. The same is applied to anything in a gravitation field, it follows the curvature of space, but since these objects have a so-called mass, we attribute a force to gravity.

Then we are talking about SR in which the curvature of space is defined as FLAT. And people like JesseM and others are saying that kinetic energy and the such contribute to an objects mass because of the force required on the body - and they refer back to gravity and GR. It is not the same thing, and any analysis of attributing a force to a moving body will show this as the definition of "relativistic mass" changes for different situations.

 

[JesseM]

I think the problem here is, we are referring to two very different things. Gravity effectively curves space which is why light bends in a gravitation field although there is no actual force on the photon. The same is applied to anything in a gravitation field, it follows the curvature of space, but since these objects have a so-called mass, we attribute a force to gravity.

Then we are talking about SR in which the curvature of space is defined as FLAT. And people like JesseM and others are saying that kinetic energy and the such contribute to an objects mass because of the force required on the body - and they refer back to gravity and GR. It is not the same thing, and any analysis of attributing a force to a moving body will show this as the definition of "relativistic mass" changes for different situations.

You don't need a detailed consideration of GR here, at least not when considering a small composite object that does not itself curve spacetime much. You can just consider the inertia of the object when you try to accelerate it in empty space using a non-gravitational force, and whatever its inertial mass in this situation, the equivalence principle tells you that this will be the same as its gravitational mass in the presence of a large object like a planet that does curve spacetime in a significant way. Or to put it another way, if the object is sitting on a scale in an elevator that is accelerating through empty space at 1G, the scale's reading should be the same as if the elevator was sitting on the surface of the earth, according to the equivalence principle.

Do you deny that in flat spacetime, SR predicts that the inertia of a compound object will be proportional to its total rest energy? Do you deny that the equivalence principle says there shouldn't be a difference between its inertial mass and its gravitational mass?

 

[Aer]

Your posts are getting more and more stupid every time you post.

You don't need a detailed consideration of GR here, at least not when considering a small composite object that does not itself curve spacetime much.

I am not considering the curvature of space by the object! Holy crap. The curvature of space is by the Earth and the object exists in this curvature.

You can just consider the inertia of the object when you try to accelerate it in empty space using a non-gravitational force, and whatever its inertial mass in this situation, the equivalence principle tells you that this will be the same as its gravitational mass in the presence of a large object like a planet that does curve spacetime in a significant way.

You must think that I don't know what the equivalence principle is and will let you get away with this retardation of the principle.

equivalence principle:states that there is no experiment a person could conduct in a small volume of space that would distinguish between a gravitational field and an equivalent uniform acceleration. This principle is the foundation of General Relavity.

your definition:the equivalence principle tells you that this [the inertia of the object when you try to accelerate it in empty space using a non-gravitational force] will be the same as its gravitational mass in the presence of a large object like a planet that does curve spacetime in a significant way.

What you may be referring to is the "weak equivalence principle" or "universality of free fall" because tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass (e.g. Dropping metal balls of different mass from the Tower of Pisa - a la Galileo).

So, effectively you are attributing a force to gravity just like I said. What is the force on a photon? You can only attribute a force to objects that already have "rest mass". You are replacing mass for acceleration in your definition of the equivalence principle and you can only do this when you do what I said above - attribute a force to gravity.

Don't even try to tell me that you know everything there possibly is to know about gravity! No scientist knows everyting there is possible to know about gravity - that is why there is debate on this issue in physics!

Back to what we do know about gravity: the acceleration is always the same, regardless of the mass of an object. In fact, an object can have no mass and will still experience the same acceleration (e.g. photons). This acceleration is due to the curvature of space, not an actual force.

Do you deny that in flat spacetime, SR predicts that the inertia of a compound object will be proportional to its total rest energy?

Of course not, if you say the total rest energy is the total rest mass form of energy.

Do you deny that the equivalence principle says there shouldn't be a difference between its inertial mass and its gravitational mass?

Like I elaborated on above, you are referring to the weak equivalence principle.

Do you deny that objects with relative motion (kinetic energy) do not behave as if they have more mass?

 

[JesseM]

Your posts are getting more and more stupid every time you post.

Again, Aer, please don't be a jerk.

I am not considering the curvature of space by the object! Holy crap.

I didn't say you were, I was just qualifying my own statement about the equivalence principle, since the argument wouldn't quite work for a very large object like a planet.

You can just consider the inertia of the object when you try to accelerate it in empty space using a non-gravitational force, and whatever its inertial mass in this situation, the equivalence principle tells you that this will be the same as its gravitational mass in the presence of a large object like a planet that does curve spacetime in a significant way.

You must think that I don't know what the equivalence principle is and will let you get away with this retardation of the principle.

equivalence principle:states that there is no experiment a person could conduct in a small volume of space that would distinguish between a gravitational field and an equivalent uniform acceleration. This principle is the foundation of General Relavity.

Seems to me that's exactly the same as my statement that "if the object is sitting on a scale in an elevator that is accelerating through empty space at 1G, the scale's reading should be the same as if the elevator was sitting on the surface of the earth, according to the equivalence principle" (assuming the elevator is considered to be a small volume of space).

your definition:the equivalence principle tells you that this [the inertia of the object when you try to accelerate it in empty space using a non-gravitational force] will be the same as its gravitational mass in the presence of a large object like a planet that does curve spacetime in a significant way.

I didn't say that this was the equivalence principle, just that it's a necessary consequence of it.

What you may be referring to is the "weak equivalence principle" or "universality of free fall" because tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass (e.g. Dropping metal balls of different mass from the Tower of Pisa - a la Galileo).

Huh? According to wikipedia the weak equivalence principle says that "The trajectory of a falling test body depends only on its initial position and velocity, and is independent of its composition." That is obviously not what I was talking about.

So, effectively you are attributing a force to gravity just like I said.

No I'm not. Read my statement about the elevator again, all I'm saying is that if the object is sitting on a scale in an elevator undergoing 1G acceleration in flat space, the reading must be the same as if the same elevator was sitting on the surface of the earth. Do you agree that an observer in a small elevator will not be able to experimentally distinguish whether he is undergoing 1G acceleration in flat space or whether he's at rest on the surface of the earth? If so, that's all you need to demonstrate that the reading on the scale will be the same, which means the inertial mass is the same as the gravitational mass.

What is the force on a photon? You can only attribute a force to objects that already have "rest mass".

Well, good thing we weren't talking about photons, we were talking about compound objects that have a rest frame. And I never said anything about treating gravity as a force, as you say, general relativity does not treat it as such.

You are replacing mass for acceleration in your definition of the equivalence principle

Again, huh? What specific quote are you referring to when you say I was "replacing mass for acceleration"?

Don't even try to tell me that you know everything there possibly is to know about gravity! No scientist knows everyting there is possibly to know about gravity - that is where there is debate on this issue in physics!

But I'm only talking about what general relativity predicts about what should happen. Anyway, most physicists expect that quantum gravity will replicate the predictions of general relativity in macroscopic domains where the spacetime curvature isn't too large.

Back to what we do know about gravity: the acceleration is always the same, regardless of the mass of an object. In fact, an object can have no mass and will still experience the same acceleration (e.g. photons). This acceleration is due to the curvature of space, not an actual force.

Sure, but different objects still have different gravitational masses, which can be measured by seeing the force they exert on a scale sitting in a gravitational field. And again, the equivalence principle shows the reading on the scale in a gravitational field must be the same as the reading on a scale in an elevator undergoing uniform acceleration (which in that case is measuring inertial mass).

Do you deny that in flat spacetime, SR predicts that the inertia of a compound object will be proportional to its total rest energy?

Of course not, if you say the total rest energy is the total rest mass form of energy.

What do you mean when you say "rest mass form of energy"? Do you agree that relativity predicts a compressed spring will have slightly more inertia than the same spring in its relaxed state, since it has a slightly larger rest energy? Do you agree that SR predicts a hot brick will have slightly more inertia than a cold one, again because it has a slightly higher rest energy?

Do you deny that the equivalence principle says there shouldn't be a difference between its inertial mass and its gravitational mass?

Like I elaborated on above, you are referring to the weak equivalence principle.

No I'm not, the quote from wikipedia shows that the weak equivalence principle is only about the trajectory of a falling object. I'm using the principle that all laws of physics should look the same in a small elevator undergoing uniform acceleration (in which a scale will measure inertial mass) as they do in the same elevator at rest in a gravitational field of equivalent strength (in which a scale will measure gravitational mass).

Do you deny that objects with relative motion (kinetic energy) do not behave as if they have more mass?

As jtbell pointed out, it's not so simple--an object in motion will be easier to accelerate in some directions then others. To avoid this issue, I'm only talking about the inertial mass of a compound, bound object in its own rest frame. In this frame, SR predicts that its inertial mass will be proportional to its total energy, and the equivalence principle predicts that its inertial mass must equal its gravitational mass. This is also noted by the authors of the paper pervect pointed to when they say "The principle of equivalence—the exact equality of inertial and gravitational mass—is a cornerstone of general relativity".

 

[Hurkyl]

What is the force on a photon?

Exactly what the definition of force says: the time derivative of momentum. Recall that an object does not need to have a nonzero rest mass in order to have nonzero momentum.

To my understanding, E=p^2c^2+m^2c^4=\gamma mc^2=m_r c^2; so yes, the m in E = mc² is relativistic mass.

You're forgetting that "E" could refer to a variety of things. For example:

E_rest = m_rest c²
E_total = γ m_rest c² = m_relativistic c²
E_kinetic = (γ - 1) m_rest c²

And, incidentally, you were looking for E² = (pc)² + (mc²)². (Where E is total energy, m is rest mass)

 

[JesseM]

She wasn't saying that's the definition: she was saying that the equivalence principle "tells you that". Since you can determine the inertia of an object by conducting an experiment in a small volume of space, she's right.

I'm a he. But otherwise, yeah, that's what I was saying--my statement wasn't supposed to be a definition of the equivalence principle, just a consequence of it.

 

[Hurkyl]

Doh, you caught it before I deleted it.

I thought I remembered you calling yourself Jessica once... apparently I was mistaken!

 

[learningphysics]

Aer, an example that has been posted by myself and others in this thread that you have not responded to:

A object heated up has greater rest mass than the same object cooled down. Do you agree with this or not?

 

[pervect]

I think the problem here is, we are referring to two very different things. Gravity effectively curves space which is why light bends in a gravitation field although there is no actual force on the photon. The same is applied to anything in a gravitation field, it follows the curvature of space, but since these objects have a so-called mass, we attribute a force to gravity.

Gravity curves/distorts space-time, not just space. It's true that one can view small objects (test masses or test light rays) as following geodesics in space-time. However, the various Christoffel symbols with time components that describe "curved"/distorted space-time can be reasonably interpreted as "forces" - for instance, the Christoffel symbol \Gamma^x{}_{tt} can be regarded as a static force in the 'x' direction, equivalent to the Newtonian gravitational force. Similarly the sum of the Christoffel symbols \Gamma^x{}_{yt}+\Gamma^x{}_{ty} can reasonably regarded as a coriolis force in the 'x' direction due to motion in the 'y' direction. (Because of symmetry concern, both of the symbols in the above sum are equal).

Light beams near a massive body curve as a result of multiple Christoffel symbols, the closest English translation to the math in my opinion is to say that part of their curvature is due to "forces" (Christoffel symbols which include time components) - the other part of the curvature of light is due to Christoffel symbols _without_ time components, which can be regraded as the curvature of _space_ (not space-time, because none of these Christoffel symbols have any time components).

The exact mathematical expression is the geodesic equation

<br /> \frac{d^2 x^i}{d \tau^2} + \Gamma^i{}_{jk} \left( \frac{dx^j}{d \tau} \right) \left( \frac{dx^k}{d \tau} \right) = 0<br />

Then we are talking about SR in which the curvature of space is defined as FLAT. And people like JesseM and others are saying that kinetic energy and the such contribute to an objects mass because of the force required on the body - and they refer back to gravity and GR. It is not the same thing, and any analysis of attributing a force to a moving body will show this as the definition of "relativistic mass" changes for different situations.

Kinetic energy does contribute to the invariant mass of a system of particles, even in SR.

Consider a closed system of particles that do not interact with the outside universe, but only with each other, and which interact with each other only when the occupy the same point in space at the same time (no fields).

Note that this simple model can be generalized to include particles that interact via fields, but making this generalization requires including the momentum and energy stored in the fields. One can alternatively model the fields as an exchange of fictitious particles.

Calculate (in geometric units where c=1) the quantity E^2 - p^2, where E is the total energy of the system of particles in some frame, and p is the total momentum of the system of particles in the same frame. You will find that this quantity is frame independent (for a _closed_ system), and is, by defintion, the invariant mass of the system.

You will find that the invariant mass of the system of particles is NOT the sum of the invariant masses of its components. The invariant mass of the system includes contributions due to the kinetic energy of the particles.

You will have to pull some clever tricks to model the equivalent of a solid sphere containing at hot gas which is _not_ expanding without the use of fields, but it's possible. You will have to devise an exchange system of fictitious particle which mimics tension to pull this off.

 

[EnumaElish]

You're forgetting that "E" could refer to a variety of things. For example:

E_rest = m_rest c²
E_total = γ m_rest c² = m_relativistic c²
E_kinetic = (γ - 1) m_rest c²

And, incidentally, you were looking for E² = (pc)² + (mc²)². (Where E is total energy, m is rest mass)

That should have been E² as you indicate. Also, the equation I was thinking about was E = m c^{2} + K = \gamma m c^2 as stated in Aer's class notes. Here m = m_rest, K is kinetic energy and \gamma={1}/{\sqrt{1-({v}/{c})^{2}}}. If one defines m_\text{relativistic} = \gamma m_\text{rest} then E_total = m_relativisticc². It can be argued that m_relativistic is not "mass," it is simply m + K/c². And if one were to define m_relativistic as "mass" then one also has to remember that unlike rest mass, m_relativistic is directional.

 

[pervect]

There are some simple SR thought experiments that can illustrate the connection between energy and "passive" gravitational mass, via the equivalence principle. The idea is to consider what sort of forces are required to counteract the "gravity" of an accelerating space-ship, and to apply the equivalence principle.

Experiment #1

Conisider a rocket accelerating with a proper acceleration equal to a. (A proper acceleration is the acceleration measured by an instantaneously co-moving observer, i.e. an obsrever in an inertial frame moving at the same speed as the rocket, with the same velocity, that is not accelerating).

Suspend a stationary particle with invariant mass m and charge q with an electric field so that it's acceleration relative to the rocket is zero. Show that m*a = q*E, where E is the electric field, i.e. E = m*a/q

----rocket----->
       m-->E

The rocket accelerates to the right. The mass m is suspended by an electric field that also points to the right, so that it does not accelerate relative to the rocket.


Experiment #2

Consider the same experiment, except that mass m is not stationary. Everything else remains the same, the rocket accelerates at the same rate, and the electric field points in the same direction as the first figure.

a) Suppose the particle is moving in a direction that's at right angles to the rocket's trajectory. The particle is moving with velocity v. Show that the electric field required to keep the particle from accelerating relative to the rocket becomes E = gamma*m*a/q, where gamma = 1/sqrt(1-(v/c)^2)

I'll omit the detailed calculations for now. People who get stuck might research "transverse mass". Perhaps someone else would like to post the detailed calculations.

b) Suppose the particle is moving in the same direction as the spaceship is accelerating with velocity v Show that the electric field required to keep the particle from accelerating relatie to the rocket is the same as case a), i.e. E=gamma*m*a/q

hint:

Relative to the comoving inertial obsrver, we require the relativistic difference v(t+dt) - a*dt = v(t) in order that the particles velocity stay constant relative to the rocket. This means that the v(t+dt) is the relativistic sum of v(t) and a*dt, i.e.

v(t+dt) = (v(t)+a*dt)/(1+v(t)*a*dt/c^2)

Show that in the limit of small dt, this implies that dv/dt = (1-(v/c)^2)*a

 

[Paulanddiw]

This is an interesting thread. When I measure my weight, I am a rest (v=0), so I'd expect my weight to be the sum of the rest masses of the particles that make up my body.

 

[pervect]

This is an interesting thread. When I measure my weight, I am a rest (v=0), so I'd expect my weight to be the sum of the rest masses of the particles that make up my body.

If you fuse two deuterium atoms to make a helium atom, the sum of masses of the two deuterium atoms is not the same as that of the resulting helium atom.

The same principle is at work with chemical binding energies and with the heat energy generated by chemical processes that is at work with nuclear binding energies and the energies (of various forms) that are generated by nuclear processes.

However, the magnitudes are a lot different - the chemical binding energies are so small that they do not make any practical difference to the mass of a body, and so are routinely ignored.

 

[pervect]

Experiment #2

Consider the same experiment, except that mass m is not stationary. Everything else remains the same, the rocket accelerates at the same rate, and the electric field points in the same direction as the first figure.

a) Suppose the particle is moving in a direction that's at right angles to the rocket's trajectory. The particle is moving with velocity v. Show that the electric field required to keep the particle from accelerating relative to the rocket becomes E = gamma*m*a/q, where gamma = 1/sqrt(1-(v/c)^2)

I'll omit the detailed calculations for now. People who get stuck might research "transverse mass". Perhaps someone else would like to post the detailed calculations.

I'll fill in the detailed calculations.

Let the direction that the particle moves in be the x direction, and let the direction that the spaceship accelerates be the y-direction.

Then the y component of the momentum (in relativistic units where c=1) in an inertial co-moving frame is

<br /> p_y = \frac{m v_y }{\sqrt{1-v_x^2 - v_y^2}}<br />

where v_x = v is the velocity of the particle in the x direction, and v_y = 0 at t=0. Because the spaceship is accelerating, v_y will be a function of time, even though its inital value is zero.

The y-component of the force on the particle is just F = qE = dp_y / dt

Now
<br /> \frac{dp_y}{dt} = \left(\frac{dp_y}{dv_y}\right)\left( \frac{dv_y}{dt}\right)<br />

But we know that d v_y/d_t = a

We can differentiate the expression of p_y as a function of v_y, and make the above substitution for d v_y/dt to get

<br /> \frac{d p_y}{dt} = \frac{m a (1-v_x^2)}{(1-v_x^2 - v_y^2)^{\frac{3}{2}}}<br />

We wish to evaluate this expression at t=0.
Substituting v_x = v and v_y = 0 and simplifying yields the final result.

F = q*E = gamma*m*a

from which E = gamma*m*a/q follows directly.