Can we deal with relativistic mass once and for all?

[PeroK]

Another problem I have is to do with the shortage of experimental evidence on macroscopic objects. So I respect what people are saying but until I can see some more evidence I shall remain a bit sceptical and keep an open mind about it. If I had to write something on SR i would probably try to keep any reference to mass to a minimum and put the emphasis on energy instead.

Not even the most ardent supporter of relativistic mass believes that more physical mass is present in one frame than another. It has never been more than a convenient (or inconvenient) notational device to simplify energy and momentum equations and make them more like their classical counterparts.

If you think relativistic mass could be the subject of active research to determine whether a) an object obeys $\vec{F} = m_R\vec{a}$

and/or

​

b) $F_g = \frac{GM_Rm_R}{r^2}$

Then, you are sadly mistaken, and have developed a fundamental misunderstanding of SR and the nature of matter.

As I said in a previous post, the concept of relativistic mass has led you astray.

 

[Orodruin]

We can have two similar objects differing in speed by 10 to minus 30 m/s or some other silly incredibly small amount

Apart from what is already said, you should note that this difference in speed is not Lorentz invariant. If you have an object A traveling at c and another object B traveling ar c - 1 m/s, the difference in speed in the rest frame of B will be c, not 1 m/s.

For small relative velocities, small changes in velocity will lead to small changes in kinetic energy. For large relative velocities you will simply lack intuition and you should therefore accept what experiments are telling you.

 

[vanhees71]

I don't think I'm being told anything new here and I'm taking everything that people are saying on board. The main point I'm trying to make is that it doesn't feel, to me, intuitively right. We can have two similar objects differing in speed by 10 to minus 30 m/s or some other silly incredibly small amount but the faster one can have billions of times more energy than the slower one. It seems to me that there's something missing. Another problem I have is to do with the shortage of experimental evidence on macroscopic objects. So I respect what people are saying but until I can see some more evidence I shall remain a bit sceptical and keep an open mind about it. If I had to write something on SR i would probably try to keep any reference to mass to a minimum and put the emphasis on energy instead.

I do not understand where your problem is. The relativistic energy (i.e., rest energy + kinetic energy) is simply given by
$$E=\frac{m_1 c^2}{\sqrt{1-v_1^2/c^2}}+\frac{m_2 c^2}{\sqrt{1-v_2^2/c^2}}.$$
For $(v/c)^2 \ll 1$ you can approximate this by
$$E \simeq (m_1+m_2) c^2 + \frac{m_1}{2} v_1^2 + \frac{m_2}{2} v_2^2.$$
Where do you have a problem with these formulae in view to your intuition? Which evidence do you exactly think is lacking?

 

[vela]

I don't think I'm being told anything new here and I'm taking everything that people are saying on board. The main point I'm trying to make is that it doesn't feel, to me, intuitively right. We can have two similar objects differing in speed by 10 to minus 30 m/s or some other silly incredibly small amount but the faster one can have billions of times more energy than the slower one. It seems to me that there's something missing.

I'd say the thing that's missing is experience with objects moving close to the speed of light relative to you. Your intuition, which is based on your everyday experience where objects move very slowly compared to the speed of light, just isn't a dependable guide here.

Another problem I have is to do with the shortage of experimental evidence on macroscopic objects. So I respect what people are saying but until I can see some more evidence I shall remain a bit sceptical and keep an open mind about it.

Do you have a reason for your skepticism other than you simply can't believe what the equations tell you? SR, after all, is a well established theory, used every day at particle accelerators all around the world. And if there were a problem with SR with macroscopic objects, you'd think this problem would reveal itself in GR as well, but it hasn't.

 

[Mister T]

This is a simple thinking about it that can avoid much of the confusion.

I agree that it's simple, but rather than avoid confusion it creates confusion, because it's not consistent with observation.

A beam of light(EM in general) has no rest mass, but a standing wave definitely have a rest mass. Thus we come to the conclusion that rest mass is nothing more than an arrested/trapped momentum(since light has a momentum as per experiment), like that in a laser cavity for example.

No. But that observation is consistent with the definition of mass as the norm of the energy-momentum 4-vector, $\sqrt{E^2-p^2}$. Thus if $E=p$ as it does for an ideal beam of light that doesn't spread, then $m=0$, but if the beam spreads then $E>p$ so $m>0$.

Energy is derived from momentum (E=integral(mv.dv)=.5 m v^2).

Apart from the fact that this is valid only in the Newtonian approximation, you have the issue that the relation can be used to derive momentum from energy. So you don't establish which is derived from the other, but rather that the two are related.

This trapping can be either by a boundary or by going in a self trapping in a circle without a boundary. Normal matter like an electron must be like that.. and of course all other matter of a higher mass. Sometimes you have a double trapping.. a trapped momentum moving with momentum trapped in something bigger. This is what you find in the nucleus. So one can say that mass and relativistic mass is momentum, and rest mass is arrested momentum.

Do you have references to support these assertions? Or are they simply an opinion rather than established physics?

It also agrees with Einstein formula E=m c^2, as the energy of a mass particle is E=.5 m v^2,

$\frac{1}{2}mv^2$ is not the correct expression for kinetic energy. It's approximately correct at low speeds, the so-called Newtonian approximation.

 

[Mister T]

Another problem I have is to do with the shortage of experimental evidence on macroscopic objects.

The GPS satellites are macroscopic objects. The atomic clocks they carry are macroscopic objects. They are designed using relativity. If relativity were incorrect the design would be flawed and you wouldn't be able to locate something using GPS.

So I respect what people are saying but until I can see some more evidence I shall remain a bit sceptical and keep an open mind about it.

That's true for all of us. But that doesn't mean we ignore evidence when it disagrees with our intuition.

If I had to write something on SR i would probably try to keep any reference to mass to a minimum and put the emphasis on energy instead.

Mass is simply the amount of energy as measured in the rest frame.

 

[stevendaryl]

Mass is simply the amount of energy as measured in the rest frame.

That makes the mass of a photon undefined, rather than zero. Maybe you can define it as a lower bound for the energy, as you measure it in different inertial frames.

 

[DrStupid]

Again, mass in the modern sense of the word (modern meaning the way one should introduce SR from the very beginning since Minkowski's ground-breaking talk/paper of 1908) is the same quantity as in non-relatistic physics, and it's a Lorentz scalar.

In another thread I demonstrated that relativistic mass is the same quantity as the mass as used in non-relativistic physics:

https://www.physicsforums.com/threads/relativistic-mass-still-a-no-no.892981/page-2#post-5620010

And relativistic mass is not mass in the modern sense of the word.

But there is M in Schwarzschild's solution and other GR solutions.

That's not gravitational mass but mass. The source of gravity in GR is the stress-energy-tensor. The fact, that this tensor may depend on mass only in special cases desn't turn mass into geravitational mass.

 

[Dadface]

Again there's been informative comments replying to my posts so thank you. I know most of it and I accept it. I already knew most of it anyway and It has all been pointed out again, including here and other threads in this forum. I'm not disputing the correctness of relativity but I can't help feeling a bit sceptical. I also sometimes feel that there's a bit more to relativity than is written about. I need to sleep on it and hopefully, with the benefit of comments made by members here, get a eureka moment. But although I accept it I have a couple of other questions.

1. In this thread people are discussing (invariant) mass. But what is it that are we discussing? What is mass and how is it defined? I know it can be defined in terms of energy but what is energy and how can that be defined? Can we get a rigorous definition of mass without going in circles where other things such as energy, work done and force etc need to be defined?

2. Is it accepted that the mass of a hydrogen atom, for example, includes the kinetic energy of the particles plus the potential energy of the particle system? If so why is it accepted that the kinetic energy of say an electron in an atom contributes to the mass of that atom but the kinetic energy of an electron outside of an atom does not contribute to the mass of the electron?

3. I think relativity would benefit by more experimental investigations. But I also think that it can't yet be done with macroscopic objects, even ping pong balls, because the technology to do so is not available as of yet. Are there any naturally occurring macroscopic objects that reach relativistic speeds that can be observed? My guess is that if there are any they exist in space - in the realm of SR.

 

[pervect]

Again, mass in the modern sense of the word (modern meaning the way one should introduce SR from the very beginning since Minkowski's ground-breaking talk/paper of 1908) is the same quantity as in non-relatistic physics, and it's a Lorentz scalar.

In another thread I demonstrated that relativistic mass is the same quantity as the mass as used in non-relativistic physics:

https://www.physicsforums.com/threads/relativistic-mass-still-a-no-no.892981/page-2#post-5620010

It's not entirely clear to me what Dr. Stupid thought he demonstrated in that thread. However, it is pretty clear to me that his formulation is not the Minkowski formulation that vanhees71 explicitly mentioned in his post.

So, if it is also clear to Dr. Stupid that hisr formulation is not the same as Miknowski's formulation, then Dr. Stupid should also agree that his post has absolutely nothing to do with what vanhees71 posted.

I would assume that Dr Stupid simply haven't yet realized yet that his formulation is not the one that Vanhees71 is talking about. Perhaps, though, he has realized this an just likes to argue? That's the other main alternative.

 

[DrStupid]

What is mass and how is it defined?

There are many ways to define mass. One of them uses the norm of the four-momentum P:

$\left| P \right|^2 = \frac{{E^2 }}{{c^2 }} - p^2 = m^2 c^2$

I know it can be defined in terms of energy but what is energy and how can that be defined?

There are many ways to define energy or momentum. One of them uses Noether’s theorem according to which the four-momentum ist he conserved property that results from homogeneity.

Can we get a rigorous definition of mass without going in circles where other things such as energy, work done and force etc need to be defined?

See above.

why is it accepted that the kinetic energy of say an electron in an atom contributes to the mass of that atom but the kinetic energy of an electron outside of an atom does not contribute to the mass of the electron?

Energy always contributes to the mass, but mass also depends on momentum (see above) and they can cancel each other out.

I think relativity would benefit by more experimental investigations.

That’s not limited to relativity. Physics always benefits from more experimental investigations.

But I also think that it can't yet be done with macroscopic objects, even ping pong balls, because the technology to do so is not available as of yet. Are there any naturally occurring macroscopic objects that reach relativistic speeds that can be observed?

Such experiments are not just a matter of speed but also of the accuracy of measurement. I wouldn’t be surprised if relativistic effects (e.g. relativistic Dopper effect) could be demonstrated with ping pong balls.

 

[Ibix]

1. In this thread people are discussing (invariant) mass. But what is it that are we discussing? What is mass and how is it defined? I know it can be defined in terms of energy but what is energy and how can that be defined? Can we get a rigorous definition of mass without going in circles where other things such as energy, work done and force etc need to be defined?

You can measure a quantity that turns out to be the four momentum by studying the recoil of a body under impact by other bodies at different speeds. The modulus of that quantity is invariant under Lorentz transform. That modulus is what we are calling mass - no more, no less. It maps well to the Newtonian concept of mass.

2. Is it accepted that the mass of a hydrogen atom, for example, includes the kinetic energy of the particles plus the potential energy of the particle system? If so why is it accepted that the kinetic energy of say an electron in an atom contributes to the mass of that atom but the kinetic energy of an electron outside of an atom does not contribute to the mass of the electron?

All the energy of the particles and fields contributes to the total mass. But the mass of the system is not the sum of the masses of its components. This is a feature of things like speed in Newtonian physics (you may be stationary, but your atoms are vibrating with rms speeds in the 100m/s range), so shouldn't be implausible.

3. I think relativity would benefit by more experimental investigations. But I also think that it can't yet be done with macroscopic objects, even ping pong balls, because the technology to do so is not available as of yet. Are there any naturally occurring macroscopic objects that reach relativistic speeds that can be observed? My guess is that if there are any they exist in space - in the realm of SR.

You may wish to read the experimental support for SR sticky thread at the top of this forum. Clocks are hardly microscopic objects.

A GPS unit isn't either, and it validates GR every day. Each LIGO detector is 4km per arm, and they're a combined unit the size of the planet. Gravity Probe B measured GR effects on a 40kg (IIRC) sphere.

 

[DrStupid]

It's not entirely clear to me what Dr. Stupid thought he demonstrated in that thread.

Just tell me what exactly remains unclear to you.

However, it is pretty clear to me that his formulation is not the Minkowski formulation that vanhees71 explicitly mentioned in his post.

So what? I just replied to the initial claim in vanhees71's post and not to the following Minkowski formulations.

 

[SiennaTheGr8]

1. In this thread people are discussing (invariant) mass. But what is it that are we discussing? What is mass and how is it defined? I know it can be defined in terms of energy but what is energy and how can that be defined? Can we get a rigorous definition of mass without going in circles where other things such as energy, work done and force etc need to be defined?

You have to bring energy into it, but that's cause for celebration! Don't you see how wonderful it is that two seemingly distinct quantities—mass and rest energy—turn out to be exactly the same thing? Energy is conserved, remember. That's why it's special. We love conserved quantities!

Einstein's $m=E_0/c^2$ is a gift. It tells us, "Don't worry about mass as its own thing anymore. It's just the rest-frame energy, and I know how much you love conserved quantities like energy. You're welcome."

(The more formal definition of mass in SR is the magnitude of the momentum four-vector, which means that $mc^2 = \sqrt{E^2 - (pc)^2}$. Note that the object is at rest when $p = 0$, in which case the total energy $E$ is the rest energy $E_0$, and so $mc^2 = E_0$.)

2. Is it accepted that the mass of a hydrogen atom, for example, includes the kinetic energy of the particles plus the potential energy of the particle system? If so why is it accepted that the kinetic energy of say an electron in an atom contributes to the mass of that atom but the kinetic energy of an electron outside of an atom does not contribute to the mass of the electron?

Forget what you think you know about mass from Newtonian mechanics. In fact, forget the word "mass" altogether and think of "rest energy" instead: the amount of energy a system has as measured in its center-of-momentum frame.

Now, say your system is a hydrogen atom. Boost to its center-of-momentum frame (its rest frame). What contributes to its total energy in this frame? Well, we've got an electron and a proton, each of which contributes its own rest energy (mass). Add to that whatever kinetic energies the particles have in this frame, and also add whatever potential energy is associated with their relative positions (it's almost entirely electromagnetic, and it's actually a negative contribution). That's all of it. The sum of these energy contributions in the atom's rest frame is the atom's rest energy (mass).

What if your system is just a single electron? Well, then the only contribution to the system's total energy in its rest frame is the electron's rest energy (its mass).

In general, you see, a system's mass isn't the sum of the masses of its constituents. This makes perfect sense when you think "rest energy"—i.e., the system's total energy as measured in its rest frame. You don't just sum up the rest energies of the constituents. You sum up all the "internal" energy contributions.

 

[Orodruin]

Einstein's $m=E_0/c^2$ is a gift. It tells us, "Don't worry about mass as its own thing anymore. It's just the rest-frame energy, and I know how much you love conserved quantities like energy. You're welcome."

I would like to double underline this. It is a fundamental insight in relativity that an object's rest energy is equal to its its inertia in its rest frame and this is the true wonder of $E_0 = mc^2$. In my relativity class, after discussing relativistic energy and momentum (i.e., 4-momentum) and 4-forces, I spend about 20 minutes on looking at the Newtonian limit with the final punchline being $m = E_0/c^2$, where $m$ was the Newtonian inertia and $E_0$ the rest-frame energy. From that point on, I start using $P^2 = m^2$ (units of $c = 1$).

 

[Mr Wolf]

Without entering into the discussion, I just want to mention two articles.
One, that I read for the first time many years ago, is: Lev B. Okun - The concept of mass (Physics today - June 1989) - If you search, you can find it online.
I also found another more recent articles by the same author: The mass versus relativistic and rest masses.
The second one is: CarI G. Adler - Does mass really depend on velocity, dad? (American Journal of Physics - January 1987).

 

[Ibix]

One, that I read for the first time many years ago, is: Lev B. Okun - The concept of mass (Physics today - June 1989) - If you search, you can find it online.

See also this thread: https://www.physicsforums.com/threads/on-the-concept-of-mass.696144/
...where several posters recommend that paper, apparently without reading the OP's user name.

 

[Mr Wolf]

Thank you! I was just reading that topic right now, searching for Okun's name here!
That article (even if it's quite old and I can't even say if it's right or no) stuck in my mind, when I studied physics, many years ago.
Reading a discussion about it is quite impressive to me.

 

[Nugatory]

The main point I'm trying to make is that it doesn't feel, to me, intuitively right. We can have two similar objects differing in speed by 10 to minus 30 m/s or some other silly incredibly small amount but the faster one can have billions of times more energy than the slower one. It seems to me that there's something missing.

What's going on here is that you've (probably without realizing it) chosen to work in a frame that tricks your intuition. Try describing the situation using at frame in which one of the two objects is a rest and and you'll have no problem intuiting why one impact crater is so much bigger than the other.

 

[atyy]

I would like to double underline this. It is a fundamental insight in relativity that an object's rest energy is equal to its its inertia in its rest frame and this is the true wonder of $E_0 = mc^2$.

Is it necessary to apply the qualifcation "in its rest frame" to "inertia" in the above statement?

 

[Jan Nebec]

Huh...this thread became one big mess and I am even more confused...so is there any general proof that inertial mass doesn't increase with velocity? Asumming only SR.

 

[Ibix]

Huh...this thread became one big mess and I am even more confused...so is there any general proof that inertial mass doesn't increase with velocity? Asumming only SR.

If you define "inertial mass" to mean "rest mass" then yes, trivially. If you define "inertial mass" to mean "relativistic mass" then no, equally trivially.

Rest mass is the better concept to use, at least in part because that's what (very nearly) everybody uses. I personally see relativistic mass as an attempt to make relativistic equations look more like Newtonian ones, which seems completely backwards to me. Relativity is not just a correction to Newton - and trying to hide the differences seems wrong-headed to me. And part of why this is causing you trouble is that you're doing the same thing - you are trying to wedge a Newtonian concept of "resistance to acceleration" into a relativistic framework where it isn't a good fit.

 

[Orodruin]

Is it necessary to apply the qualifcation "in its rest frame" to "inertia" in the above statement?

If you want to use the Newtonian concept of inertia and of force being the time derivative of momentum at the same time as you have an equality and not just a very good approximation I would say so, yes.

 

[sweet springs]

I would add the comment, though it might be already mentioned in the thread, that in any inertial frame of reference energy and momentum of the system satisfy

$$E^2-P^2c^2=E'^2-P'^2c^2=E"^2-P"^2c^2=...=Constant$$

Square root of this positive constant is worth named as rest energy or (rest) mass when divided by c^2.

Opposite to OP's idea, we may do not use the word (rest) mass at all replacing it by rest energy. Thus relativistic mass in any frame of reference simply means energy in that frame divided by c^2. We can save the word mass.

 

[atyy]

Huh...this thread became one big mess and I am even more confused...so is there any general proof that inertial mass doesn't increase with velocity? Asumming only SR.

There is more than one definition of the term "inertia" or "mass". All of them are correct.

1) If you define "mass" as the thing in Newton's second law "F=dp/dt", then you can use the relativistic mass to get the correct relativistic law for the 3-force. http://www.feynmanlectures.caltech.edu/I_15.html

2) If you define "mass" as the thing in Newton's second law "F=ma", then you can use the longitudinal and transverse masses to get the correct relativistic law for the 3-force. https://www.amazon.com/dp/0198567324/?tag=pfamazon01-20 (Eq 6.51)

3) If you define "mass" as the thing in Newton's second law "F=ma", then you can use the invariant mass (also known as the rest mass) to get the correct relativistic law for the 4-force. https://ocw.mit.edu/courses/physics/8-033-relativity-fall-2006/readings/dynamics.pdf

 

[atyy]

If you want to use the Newtonian concept of inertia and of force being the time derivative of momentum at the same time as you have an equality and not just a very good approximation I would say so, yes.

So is that some form of "relativistic mass"?

 

[Orodruin]

So is that some form of "relativistic mass"?

No. I do not see how you can come to that conclusion based on what I have said.

 

[SiennaTheGr8]

If you want to use the Newtonian concept of inertia and of force being the time derivative of momentum at the same time as you have an equality and not just a very good approximation I would say so, yes.

So is that some form of "relativistic mass"?

No, he's saying that the relativistic relationship between 3-force and 3-acceleration "reduces" to the Newtonian relation only in the rest frame. In general (for constant $m$), we have:

$\vec f = \gamma^3(\vec v \cdot \vec a) m \vec v + \gamma m \vec a$.

Only in the rest frame (where $v=0$ and $\gamma=1$) does this reduce to:

$\vec f = m \vec a$.

In this special frame, and in this special frame only, we can unambiguously point to $m$ [$=E_0 / c^2$] as the "measure of inertia."

 

[sweet springs]

In Newtonian mechanics momentum is expressed

$$\mathbf{P}=m\mathbf{v}$$

Einstein find the correction

$$\mathbf{P}=m\frac{1}{1-\sqrt\frac{v^2}{c^2}}\mathbf{v}$$

There are two ways of interpretation

$$\mathbf{P}=(m\frac{1}{1-\sqrt\frac{v^2}{c^2}})\mathbf{v}=m(\mathbf{v})\mathbf{v}$$
rerativistic mass.

$$\mathbf{P}=m(\frac{1}{1-\sqrt\frac{v^2}{c^2}}\mathbf{v})=m\ c\mathbf{u(v)}$$
four-velocity, the spatial components of.

The latter is simple because u is the only function of v.

$$\vec f = \gamma^3(\vec v \cdot \vec a) m \vec v + \gamma m \vec a.$$

well shows bothersome of the former interpretation.

 

[atyy]

No. I do not see how you can come to that conclusion based on what I have said.

Is there an expression for the inertia of an object that is not in its rest frame?

 

[PAllen]

Is there an expression for the inertia of an object that is not in its rest frame?

[edit: corrections in line below]
Depends how you are willing to define inertia. If you take momentum to be mass * celerity, and force as time derivative of momentum, then for the case of no flows of matter or energy into/out of the body, you have force = mass * (rate of change of celerity). Note that for slow speeds, celerity goes to velocity, and change of celerity goes to acceleration. If you accept that these things are the relativistic generalizations for velocity and acceleration, then it makes perfect sense to consider invariant mass the inertia in all cases. Invariant mass is simply resistance to change of celerity.

Note also that magnitude of rate of change of celerity equals proper acceleration of the body [when acceleration and velocity are in the same direction], further justifying this interpretation. Gedanken experiment : attach accelerometer to body, apply 3-force, in any frame [in the direction of velocity], and the ratio is invariant mass = inertia.

 

[Orodruin]

Is there an expression for the inertia of an object that is not in its rest frame?

To add to what PAllen said, I was discussing the Newtonian limit and how you conclude that the rest energy is the same thing as the Newtonian inertial mass. In that setting there is no reason to look at anything but $v \to 0$.

 

[atyy]

To add to what PAllen said, I was discussing the Newtonian limit and how you conclude that the rest energy is the same thing as the Newtonian inertial mass. In that setting there is no reason to look at anything but $v \to 0$.

So are you considering that v is not zero before taking the limit?

Just to clarify, I don't understand why "in its rest frame" had to be added as a qualification. If one is talking about either the rest energy or the rest mass, then that is a frame-independent invariant.

 

[Orodruin]

So are you considering that v is not zero before taking the limit?

Just to clarify, I don't understand why "in its rest frame" had to be added as a qualification. If one is talking about either the rest energy or the rest mass, then that is a frame-independent invariant.

The point is I am not talking about the rest energy or rest mass (on one side of the equation). Those are relativistic concepts. I am talking about the equivalence of rest energy on the left-hand side to the notion of Newtonian inertia on the right-hand side. This is a non-trivial result that is often haphazardly glossed over by introducing the invariant mass as the square of the 4-momentum and then just assuming there is an equivalence. It is the reason we call the rest energy a ”mass” in the first place.

In any frame other than the rest frame, the Newtonian concept of inertia is incompatible with the relativistic relations at some level.

 

[atyy]

The point is I am not talking about the rest energy or rest mass (on one side of the equation). Those are relativistic concepts. I am talking about the equivalence of rest energy on the left-hand side to the notion of Newtonian inertia on the right-hand side. This is a non-trivial result that is often haphazardly glossed over by introducing the invariant mass as the square of the 4-momentum and then just assuming there is an equivalence. It is the reason we call the rest energy a ”mass” in the first place.

In any frame other than the rest frame, the Newtonian concept of inertia is incompatible with the relativistic relations at some level.

Ok, got it. So there is no relativistic notion of inertia in your conception.