Weight of a relativistic particle

[superdave]

If I were able to accelerate a particle to relativistic speeds, then capture that particle in a box in such a way that it kept its speed, then weighed that box, would it weigh more than the box + rest weight of the particle? Would it exert a gravitational field greater than that of the box with the particle at rest inside?

 

[phinds]

If I were able to accelerate a particle to relativistic speeds, then capture that particle in a box in such a way that it kept its speed, then weighed that box, would it weigh more than the box + rest weight of the particle? Would it exert a gravitational field greater than that of the box with the particle at rest inside?

You are basically asking whether or not the deprecated term "relativistic mass" is the same as rest mass. It is not and the REASON it's a deprecated term is that it causes exactly this kind of confusion. Mass is an inherent property. The stress energy tensor in General Relativity ("realtivistic mass") is not; it is a function of relative speed.

If your particle is traveling at relativistic speeds in one FOR and another particle is traveling at the same speed in the same FOR and parallel to the first, then the two have only "rest mass" attraction to each other because they have zero relative velocity. If they pass by a particle that is at rest in the FOR, then the two moving particles and the rest particle have a greater attraction to each other than would be explained by rest mass alone.

 

[superdave]

You are basically asking whether or not the deprecated term "relativistic mass" is the same as rest mass.

Well, more specifically, I'm asking whether or not the effect that was once referred to as a the deprecated term "relativistic mass" is indistinguishable from rest mass in certain situations such as the one I described above.

I understand that there are legitimate reasons for defining rest mass a certain way, and the relativistic effects another. But are there situations where, at a small, short scale, the relativistic effects are identical to replacing it with an object with larger rest mass?

 

[Dale]

If I were able to accelerate a particle to relativistic speeds, then capture that particle in a box in such a way that it kept its speed, then weighed that box, would it weigh more than the box + rest weight of the particle? Would it exert a gravitational field greater than that of the box with the particle at rest inside?

A box with a hot gas will weigh more than a box with the same gas but cold.

 

[superdave]

A box with a hot gas will weigh more than a box with the same gas but cold.

And assuming the box is a perfect thermal insulator (I guess sealed after the gas was heated), is there anyway to tell whether it is a box of x hot gas particles or y cold gas particles?

 

[Dale]

And assuming the box is a perfect thermal insulator (I guess sealed after the gas was heated), is there anyway to tell whether it is a box of x hot gas particles or y cold gas particles?

I am sure there are several ways, but the one that comes to mind immediately is measuring sound wave velocity in the box material. Or even just a strain gauge. You might even be able to measure sound wave velocity in the gas. In principle you might be able to detect collisions of individual gas molecules. Maybe radar. Or depending on the gas you could use MRI, diffusion MRI would work well. I don’t know if there are any gasses that emit positrons or gamma rays. OK, so more than one way came to mind with a little thought.

 

[_PJ_]

I understand that there are legitimate reasons for defining rest mass a certain way, and the relativistic effects another. But are there situations where, at a small, short scale, the relativistic effects are identical to replacing it with an object with larger rest mass?

You could rearrange to work in momentum, this way you can even include massless objects

 

[Mister T]

If I were able to accelerate a particle to relativistic speeds, then capture that particle in a box in such a way that it kept its speed, then weighed that box, would it weigh more than the box + rest weight of the particle?

If you have a box moving at a high speed, with a particle inside also moving at the same fast speed and in the same direction you would have exactly the same situation as if you had a box at rest with a particle inside it also at rest. This is the Principle of Relativity. A state of rest is equivalent to a state of uniform motion.

On the other hand, if you have a box of mass ##M## with a particle of mass ##m## inside the box, moving relative to the box then the mass of this system would be more than ##M+m##. This is true if the box is at rest relative to you, or moving very fast relative to you.

 

[pervect]

If you have a box moving at a high speed, with a particle inside also moving at the same fast speed and in the same direction you would have exactly the same situation as if you had a box at rest with a particle inside it also at rest. This is the Principle of Relativity. A state of rest is equivalent to a state of uniform motion.

On the other hand, if you have a box of mass ##M## with a particle of mass ##m## inside the box, moving relative to the box then the mass of this system would be more than ##M+m##. This is true if the box is at rest relative to you, or moving very fast relative to you.

A moving box (also a moving particle) would weigh more, as weighted by a scale. Techniques for weighting moving objects are known (for instance there are commercial systems that weigh moving trucks), though not at relativistic speeds. Most are based on measuring pressure on a sensor plate, I believe.

I looked up a few old threads on the topic, but I didn't find one that I'd particularly care to recommend :(. What happens though is that the total force on the moving block increases by ##\gamma##.

[add]
A wiki reference is <<link>>:

The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass

moving in the x direction with velocity v and associated Lorentz factor

is

f x = m γ 3 a x = m L a x , f y = m γ a y = m T a y , f z = m γ a z = m T a z .

##\gamma m## is often called the transverse mass.

 

[Mister T]

A moving box (also a moving particle) would weigh more, as weighted by a scale.

What is the definition of weight that you are using here? You do realize that more than just one or two are in common usage, that's within the physics community and outside of it.

Techniques for weighting moving objects are known (for instance there are commercial systems that weigh moving trucks), though not at relativistic speeds. Most are based on measuring pressure on a sensor plate, I believe.

Using the definition of weight used by the commercial trucking industry, they would not. I think you mean that the force the truck exerts on the platform is different when the truck is moving? Some people in the physics community do use that as their definition of weight, although they usually take to calling it the apparent weight. Most people within the physics community would call that the normal force. In many cases outside the physics community, for example in the commercial trucking industry, the definition of weight is precisely (by law) equivalent to what people in the physics community, and indeed all of science, call mass.

Say you have a truck with a gross weight of 4567 tons, measured when the truck is at rest on the typical platform scale used at road side weigh stations. Those commercial scales you speak of would have to be calibrated so they read 4567 tons when the truck moves across them. Otherwise those responsible would be in violation of the law.

I looked up a few old threads on the topic, but I didn't find one that I'd particularly care to recommend :(. What happens though is that the total force on the moving block increases by ##\gamma##.

Again, this would depend on the definition being used for weight. Even within the physics community where it's virtually always insisted that weight be defined as a force, physicists cannot agree on how to define that force. I have references to support that. Some of them are particularly poignant.

 

[pervect]

What is the definition of weight that you are using here? You do realize that more than just one or two are in common usage, that's within the physics community and outside of it.
Using the definition of weight used by the commercial trucking industry, they would not. I think you mean that the force the truck exerts on the platform is different when the truck is moving? Some people in the physics community do use that as their definition of weight, although they usually take to calling it the apparent weight. Most people within the physics community would call that the normal force. In many cases outside the physics community, for example in the commercial trucking industry, the definition of weight is precisely (by law) equivalent to what people in the physics community, and indeed all of science, call mass.

Say you have a truck with a gross weight of 4567 tons, measured when the truck is at rest on the typical platform scale used at road side weigh stations. Those commercial scales you speak of would have to be calibrated so they read 4567 tons when the truck moves across them. Otherwise those responsible would be in violation of the law.
Again, this would depend on the definition being used for weight. Even within the physics community where it's virtually always insisted that weight be defined as a force, physicists cannot agree on how to define that force. I have references to support that. Some of them are particularly poignant.

The point I'm trying to make is that the response of such a scale would depend on the velocity of the truck. The same scale, without recallibration, would read a higher number for a relativistically moving truck than a stationary truck. I think you're right about the legalities, the legal definition of weight seems to be different than the physics definition. The sort of weight one measures with a scale isn't the sort of weight we use in commerce. I'm not sure why the physics defintion is what it is, but that is my understanding of the common usage of weight as it is used in physics and engineering, where we measure weight (and force) in Newtons.

I'm not sure what papers you refer to that argue about issues about how to define the force. At the most basic level, I would treat the problem as a point particle (which avoids some interesting but advanced physics). I'm not aware of any issues with defining the force on a point particle, if you think you have some references that call this into question, I'd like to see them if you think the point is worth discussing. (We could discuss the non-point particle case if that's where the objection is too, but the discussion will become more advanced and less clearcut.)

The wiki quote I added to my original post summarizes the simplest I level treatment I know of. As wiki mentions, the ratio of the force to the rest mass increases for a relativistically moving particle. The rest mass of a particle doesn't change when it moves (by definition). The ratio of force/accelerationdoes change when the particle moves, and it changes differently for when the particle moves transverse to the force (the so-called transverse mass) to when the particle moves in the direction of the force (the so called longitudinal mass).

I actually don't use the concept of "transverse mass" much, but it serves as a convenient way to find documented calculations of the force in this case.

 

[JulianM]

Why is "relativistic mass" "deprecated"? since it is a direct result of SR?

 

[phinds]

Why is "relativistic mass" "deprecated"? since it is a direct result of SR?

see post #2

 

[Dale]

Why is "relativistic mass" "deprecated"? since it is a direct result of SR?

see: https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

 

[Mister T]

Why is "relativistic mass" "deprecated"? since it is a direct result of SR?

It's not a consequence of the postulates. Instead it's simply a definition.

 

[JulianM]

see post #2

Post #2 is really not very clear to me.

Decoded it says the reason is that we express disapproval of it becatuse it causes confusion and that the confusion originates from the fact that 2 objects traveling at the same speed which is a relativistic speed (hard to understand) then they are at rest relative to each other - well ok.

Next it says that another object passes them at a relativistic speed in the same frame of reference, which also doesn't seem to make sense, will cause an attraction. This doesn't say anything really about mass.

Now taking the OP and applying post #2 where an object passes the box and weighing scale then the speed of the passing object will create an attraction, resulting in a higher reading on the scale but we don't say this is an increase in mass, but rather an attraction between objects.

How do we derive the attractive force from the speed of the passing object?

 

[phinds]

Post #2 is really not very clear to me.

Decoded it says the reason is that we express disapproval of it becatuse it causes confusion and that the confusion originates from the fact that 2 objects traveling at the same speed which is a relativistic speed (hard to understand) then they are at rest relative to each other - well ok.

Next it says that another object passes them at a relativistic speed in the same frame of reference, which also doesn't seem to make sense, will cause an attraction. This doesn't say anything really about mass.

Now taking the OP and applying post #2 where an object passes the box and weighing scale then the speed of the passing object will create an attraction, resulting in a higher reading on the scale but we don't say this is an increase in mass, but rather an attraction between objects.

How do we derive the attractive force from the speed of the passing object?

As stated in post #2, "Mass is an inherent property. The stress energy tensor in General Relativity ("realtivistic mass") is not; it is a function of relative speed."

 

[Ibix]

Post #2 is really not very clear to me.

I think relativistic mass was introduced to make some relativistic equations "look Newtonian". For example ##p=\gamma m_0v## reduces to the familiar Newtonian momentum expression if you define ##m=\gamma m_0##.

This has always seemed backwards to me - disguising the more precise theory to make it look like the less precise one. And students immediately assume that relativity is just Newton with a variable ##m##, which isn't remotely true - you can't just plug ##\gamma m## into ##F=GMm/r^2##, for example.

Relativistic mass is just the total energy of a particle, since ##E=\gamma mc^2## (for a massive particle). And since "total energy" also makes sense for massless particles ("yes, massless particles have non-zero relativistic mass!" ) it's a better term.

 

[JulianM]

As stated in post #2, "Mass is an inherent property. The stress energy tensor in General Relativity ("realtivistic mass") is not; it is a function of relative speed."

Splitting this into it's two parts then:

- in SR mass is considered invariant i.e. an object does not increase in mass

- in GR (which was not the context of the discussion, though) there is a change in mass which post #2 caused by an attractive force.

 

[phinds]

... in GR (which was not the context of the discussion, though) there is a change in mass which post #2 caused by an attractive force.

No, it's not an attractive force. Gravity is not a force in GR, it's just space-time geometry.

 

[JulianM]

I was trying to decipher the meaning of "If they pass by a particle that is at rest in the FOR, then the two moving particles and the rest particle have a greater attraction to each other than would be explained by rest mass alone"

 

[phinds]

I was trying to decipher the meaning of "If they pass by a particle that is at rest in the FOR, then the two moving particles and the rest particle have a greater attraction to each other than would be explained by rest mass alone"

As indeed they would, regardless of whether they are massless or massive particles, because their relative velocity figures into the stress energy tensor which gives what can be called "effective total mass". [NOTE: of course, if they were massless particles there is no FOR in which any of them could be at rest]

 

[JulianM]

This is jumping around a little, I thought you were saying that the stress energy tensor was derived from GR.

The OP related to SR and was in reference to a massive particle so we don't need to consider massless particles at this stage.

So in post #2 we have two inertial objects traveling at some relativistic speed and then in the same frame of reference an object at rest and that the various masses do not vary as a result of relative velocities.

The meaning of that in this scenario would therefore be the mass of all the objects is invariant and that we deprecate (disapprove of) the term "relativistic mass" because the mass is invariant?

 

[PeroK]

The meaning of that in this scenario would therefore be the mass of all the objects is invariant and that we deprecate (disapprove of) the term "relativistic mass" because the mass is invariant?

Things are what you define them to be. If you want to define the mass of a particle as ##\gamma m## and thus make it frame dependent, you are free to do that. If you want to define the mass of a particle as its rest mass, ##m##, you are free to do that as well. But, you must ensure that all your formulas and working reflect your chosen definition.

On balance, defining mass as ##\gamma m## has more disadvantages than advantages, so has been generally dropped from modern physics texts.

The idea that there is a deep physical truth lurking in whether mass should be ##\gamma m## or ##m## is misguided. Mass is what we define it to be.

Gravitation in GR is not just the result of mass, nor of energy, nor of energy density and energy flux, but also of the stress tensor. This "stress-energy" tensor is a second order tensor with, on the face of it, 16 terms. That is a very different beast from relativistic mass.

 

[JulianM]

Things are what you define them to be. If you want to define the mass of a particle as ##\gamma m## and thus make it frame dependent, you are free to do that. If you want to define the mass of a particle as its rest mass, ##m##, you are free to do that as well. But, you must ensure that all your formulas and working reflect your chosen definition.

On balance, defining mass as ##\gamma m## has more disadvantages than advantages, so has been generally dropped from modern physics texts.

The idea that there is a deep physical truth lurking in whether mass should be ##\gamma m## or ##m## is misguided. Mass is what we define it to be.

Gravitation in GR is not just the result of mass, nor of energy, nor of energy density and energy flux, but also of the stress tensor. This "stress-energy" tensor is a second order tensor with, on the face of it, 16 terms. That is a very different beast from relativistic mass.

I am not confusing mass with weight. Weight is a measure of the force of gravity. Mass is a measure of the amount of "stuff" in the object, as we know, so gravity is not a concern.

So help me to understand why I can define mass as "anything I want it to be". I understand i can use various units, of course, but stuff is still stuff, isn't it?

 

[PeroK]

I am not confusing mass with weight. Weight is a measure of the force of gravity. Mass is a measure of the amount of "stuff" in the object, as we know, so gravity is not a concern.

So help me to understand why I can define mass as "anything I want it to be". I understand i can use various units, of course, but stuff is still stuff, isn't it?

If you take the "stuff is stuff" philosophy, then you can't have any time for relativistic mass, which gives a different mass in every reference frame. For example, if you accelerate away from an object, creating a relative velocity between you and the object, how could that create more "stuff" in the object? Especially, as someone who remains at rest relative to the object sees no change in it.

So, there's your answer about why you prefer not to use relativistic mass.

But, defining mass as ##\gamma m## is perfectly tenable. Many people used to do that - and some still do!

 

[JulianM]

So I can agree with your statement "if you accelerate away from an object, creating a relative velocity between you and the object, how could that create more "stuff""

except that we are still dealing with inertial frames, not acceleration, but I get your meaning.

Now gamma is the Lorentz factor which contains the velocity v. When v gets large then gamma gets large so we have to conclude that defining mass as gamma.mass implies that it's mass varies according to its velocity relative to something.

Now since its gamma.mass is dependent on its relative velocity (to something) why do we "deprecate" (disapprove of) the term relativistic mass. Doesn't the Lorentz factor tell us that it is relative?

 

[Ibix]

Now since its gamma.mass is dependent on its relative velocity (to something) why do we "deprecate" (disapprove of) the term relativistic mass.

See post #18.

Doesn't the Lorentz factor tell us that it is relative?

Depends if by "mass" you mean "rest mass" or "relativistic mass". Most modern sources mean "rest mass"; opinion is divided in older sources. The modern view seems more sensible to me, and it's always a good idea to use standardised terminology where available.

 

[PeroK]

Now since its gamma.mass is dependent on its relative velocity (to something) why do we "deprecate" (disapprove of) the term relativistic mass. Doesn't the Lorentz factor tell us that it is relative?

Okay, then you would prefer to use relativistic mass? It seems more logical to you? But, then, where is your "stuff is stuff" idea?

It's got to be one or the other. You can't have both. Although, I guess you could. On Mondays, Wednesdays and Fridays, you could use the invariant mass ##m##. And, on Tuesdays, Thursdays and Saturdays you could use relativistic mass ##\gamma m##.

And, if someone asks you why, you could say: I like the idea that "stuff is stuff" so I like the idea of invariant mass. But, I also like the fact that mass changes with relative velocity, so I like relativistic mass too. So, I use one some days and the other on other days.

Physicists decided they needed to settle on one or the other, and in general they preferred invariant mass. Relativistic mass is deprecated because that is people not following an agreed standard.

 

[SiennaTheGr8]

I am not confusing mass with weight. Weight is a measure of the force of gravity. Mass is a measure of the amount of "stuff" in the object, as we know, so gravity is not a concern.

So help me to understand why I can define mass as "anything I want it to be". I understand i can use various units, of course, but stuff is still stuff, isn't it?

Mass is not "amount of stuff." Even in Newtonian mechanics it isn't, though there you can more or less get away with thinking about mass in that way, as a heuristic. But really Newtonian mass is the resistance to acceleration / the constant of inverse-proportionality between density and volume / the gravitational "charge."

In special relativity, mass is a measure of how much energy a system has as measured in its center-of-momentum frame (its rest energy). That includes kinetic- and potential-energy contributions "internal" to the system—i.e., the system's mass isn't just the sum of the masses of the system's constituents—so it's no longer possible to use the "amount of stuff" heuristic at all. Mass in SR isn't even the resistance to acceleration!

 

[PeroK]

Mass in SR isn't even the resistance to acceleration!

You have ##\bf{f}## ##= m \bf{a}##

 

[SiennaTheGr8]

You have ##\bf{f}## ##= m \bf{a}##

Only in the instantaneous rest frame.

(Unless those are meant to be four-vectors.)

 

[PeroK]

Only in the instantaneous rest frame.

(Unless those are meant to be four-vectors.)

What else would they be?

 

[SiennaTheGr8]

What else would they be?

Lowercase boldface is usually used for three-vectors in my experience.

 

[PeroK]

Lowercase boldface is usually used for three-vectors in my experience.

What would you use for the four-velocity?