Understanding Mass in Relativity Theory: Definitions and Conflicting Views

[Dadface]

Most of the current introductory textbooks that I have at hand (for college/university level in the US) simply use equations that are written in terms of invariant mass (your m₀, but usually simply called m). That is, they write e.g. $p = mv / \sqrt{1 - v^2/c^2} = \gamma mv$ instead of p = mv. They do not mention the so-called "relativistic mass" at all, except sometimes as a historical footnote for the benefit of students who have seen it elsewhere.

The only exception in my admittedly small collection is French's "Newtonian Mechanics" which I think is still somewhat popular even though it was written over forty years ago.

Thank you jtbell. Let me refer to my equation again without the subscript and exchanging E+m for M:

E+m=mL

The way I understand the equation is that m stands for the invariant mass (rest mass) and E stands for the kinetic energy expressed in mass units. I understand from yourself and others that the equation fell out of favour. Is it because the equation is considered to be incorrect? Is it not a good equation for calculating the kinetic energ of a body of known invariant mass and known velocity?
Another thing that concerns me is how does SR deal with those problems where the equation assumes the existence of an invariant mass but where the mass is not,in fact, invariant?

 

[Phy_Man]

Conservation of mass/energy does not apply in GR...that is, in curved spacetime.

There is no universal frame to even define velocity...or time...precisely...

That is incorrect. One doesn't need a universal frame to express the conservation of energy. One uses the differential equation of continuity to express the local conservation of energy-momentum. The expression for local conservtion of energy-momentum is given by

$$\nabla_{\mu}T^{\mu\nu} = 0$$

You can find this expression in any decent GR text.

 

[Dadface]

Conservation of mass/energy does not apply in GR...that is, in curved spacetime.

There is no universal frame to even define velocity...or time...precisely...

Anyway, you don't need a lot of mass to form a black hole... Gravitational collapse occurs when an object's internal pressure is insufficient to resist the object's own gravity. That could be a pea sized mass...or smaller...

Basically I am interested in the validity or otherwise of the apparently discredited equation I referred to in my earlier post. if GR does not accommodate the conservation of mass energy is that a limitation of GR or of SR or of something else or does it suggest that mass/energy is not conserved?

Also, if there is no universal frame to define velocity or time precisely then is it not so that any limitations this imposes apply to all relevant equations of SR and not just the equation I referred to?

 

[WannabeNewton]

Local energy conservation always holds in GR. Global energy conservation will hold if the space-time has a generator of time translation symmetry, which is true for e.g. Minkowski space-time (which is what SR is set on).

 

[Naty1]

This all sounds good...
On the Abuse and Use of Relativistic Mass by Gary Oas

Abstract - The concept of velocity dependent mass, relativistic mass, is examined and is found to be inconsistent with the geometrical formulation of special relativity. ... It is argued that the oft-held view that formulations of relativity with and without relativistic mass are equivalent is incorrect. ... As geometry lies at the heart of all modern representations of relativity, it is urged, once again, that the use of the concept at all levels be abandoned.

What does this mean:

Physicists are always ignoring the stress-energy-momentum tensor in special relativity...

The former is the source for gravitational spacetime curvature; in SR there is no gravitational spacetime curvature...it's flat spacetime...

 

[dextercioby]

Special relativity includes classical fields on flat background in which the 4-momentum of the field comes from the stress-energy 4-tensor, so the statement >Physicists are always ignoring the stress-energy-momentum tensor in special relativity...< is incorrect.

 

[Phy_Man]

What does this mean:

The former is the source for gravitational spacetime curvature; in SR there is no gravitational spacetime curvature...it's flat spacetime...

While it is correct that the stress-energy-momentum tensor is the source of gravity it is incorrect to think that is its only purpose. For example; if you wanted to find the momentum of an arbitrary distribution of matter then you'd have to use the stress-energy-momentum tensor. Rindler gives a good example of this in his SR/GR/Cosmology text. The example he uses first appeared in the article A simple relativistic paradox about electrostatic energy by Wolfgang Rindler and Jack Denur, Am. J. Phys. 56, 795 (1988)

Abstract - A charged parallel‐plate vacuum capacitor moves uniformly through an inertial frame. Its field energy alone does not transform according to the familiar law ‘‘energy=γ× rest energy.’’ However, when the stresses in the supports are taken into account, the entire system does satisfy this relation.

It's important to understand that stress/pressure has inertia. Schutz explains it in his text Gravity from the Ground Up. The following article is very interesting - The inertia of stress by Rodrigo Medina, Am. J. Phys. 74, 1031 (2006)

Abstract - We present a simple example in which the importance of the inertial effects of stress is evident. The system is an insulating solid narrow disc whose faces are uniformly charged with charges of equal magnitude and opposite signs. The motion of the system in two different directions is considered. It is shown how the contributions to energy and momentum of the stress that develops inside the solid to balance the electrostatic forces have to be added to the electromagnetic contributions to obtain the results predicted by the relativistic equivalence of mass and energy.

A good understanding of the stress-energy-momentum tensor is very enlightening. I highly recommend learning everything you can about it. Tolman's text explains it pretty well.

Special relativity includes classical fields on flat background in which the 4-momentum of the field comes from the stress-energy 4-tensor, so the statement >Physicists are always ignoring the stress-energy-momentum tensor in special relativity...< is incorrect.

I know. I was exagerating. What I literally meant was that most SR texts ignore that tensor. It'd be like me saying "My wife is always on the phone." Nobody would take me literally. I shouldn't have made it seem that way though. Therefore thanks for pointing that out.

 

[WannabeNewton]

What does this mean:

The former is the source for gravitational spacetime curvature; in SR there is no gravitational spacetime curvature...it's flat spacetime...

It's an incorrect statement. Even in SR we have electromagnetic fields, Klein Gordon fields etc. propagating on the background flat space-time and they have an associated energy-momentum tensor; we just use $\eta_{ab}$ and the associated derivative operator $\partial_{a}$. For example, the energy-momentum tensor of a Klein Gordon in SR is $T_{ab} = \partial_{a}\varphi \partial_{b} \varphi - \frac{1}{2}\eta_{ab}(\partial^{c} \varphi \partial_{c}\varphi + m^{2}\varphi^{2})$. So physicists aren't ignoring it; any decent text/introductory notes on classical field theory will cover the energy-momentum tensor in SR.

 

[Phy_Man]

It's an incorrect statement. Even in SR we have electromagnetic fields, Klein Gordon fields etc. propagating on the background flat space-time and they have an associated energy-momentum tensor; we just use $\eta_{ab}$ and the associated derivative operator $\partial_{a}$. For example, the energy-momentum tensor of a Klein Gordon in SR is $T_{ab} = \partial_{a}\varphi \partial_{b} \varphi - \frac{1}{2}\eta_{ab}(\partial^{c} \varphi \partial_{c}\varphi + m^{2}\varphi^{2})$. So physicists aren't ignoring it; any decent text/introductory notes on classical field theory will cover the energy-momentum tensor in SR.

Huh? I already addressed this in the post right before this one. As I said, that's not what I was referring to and it wasn't meant to be taken literally. I was talking about things like the inertia of stress. I never intended to imply that the stress-energy-momentum tensor can't be found in texts which use special relativity. I was referring to texbooks which teach special relativity. I went back and modifed that post so people don't keep reading it other than I intended.

 

[WannabeNewton]

https://www.amazon.com/dp/1852334266/?tag=pfamazon01-20
Here's the textbook I used to learn SR; it goes into all the aforementioned material (except KG fields).

 

[Phy_Man]

https://www.amazon.com/dp/1852334266/?tag=pfamazon01-20

And I also wrote A good understanding of the stress-energy-momentum tensor is very enlightening. I highly recommend learning everything you can about it. Tolman's text explains it pretty well. Just because it also covers GR don't take that to mean it's not a great SR text.

 

[WannabeNewton]

Since I just got through explaining that what I said was that SR texts often ignore the stress-energy-momentum tensor the purpose of this post was ...

Again you are just making blanket statements. Anyone can make up random statistics on the spot. There are a myriad of counter examples. Just because you haven't seen them doesn't mean they don't exist.

 

[Phy_Man]

It's an incorrect statement. ...So physicists aren't ignoring it; any decent text/introductory notes on classical field theory will cover the energy-momentum tensor in SR.

I got to ask. Did you really believe that I was actually claiming to know what every single physicist in the world ignores and doesn't ignore? :tongue:

 

[Phy_Man]

Again you are just making blanket statements. Anyone can make up random statistics on the spot. There are a myriad of counter examples. Just because you haven't seen them doesn't mean they don't exist.

Take a look at the content of the posts in this thread and ask yourself if each person actually did a count of what physicists use and don't use. You really shouldn't bother with comments like this. Nobody ever assumes that when someone makes a comment about what is and isn't used that they actually went to the library of Congress and did a book count or went through decades of journals and counted the number of times a quantity appeared and didn't appear. When people make statements like that you should know what it means and not take these things literally or assume that they were looking at a statistical guide. Its a waste of bandwidth and everyones time to determine what's meant to be taken litterally or not.

 

[Phy_Man]

Again you are just making blanket statements. Anyone can make up random statistics on the spot. There are a myriad of counter examples. Just because you haven't seen them doesn't mean they don't exist.

I can't believe that you actually assumed that what I posted was meant to be taken as the literal truth. If you actually believed that I was claiming to know what every single SR textbook ever printed contains then how can people take what you say seriously? If you're unable to tell when someone exagerating then you should ask them, or at least read the posts which followed where it was already stated that they were exagerating.

I really have to question the reason you posted all of this when I clearly stated in the post before yours that I was exagerating like someone who says "My wife never gets off the phone" is exagerating. I never said I made up statistics. But I've been studying relativity for some 30 years now and have a good feeling about what's out there and know that what I said is not literally true.

Next time just ask or read what has already been posted. After all you didn't accuse yechin of using blanket statistics when in post #4 in this thread he wrote ...nowadays most people use... regarding his comment about what "most" people use. You only did it to me. Why you singled me out we'll never know.

 

[Dadface]

It's me again. I refer once more to the relativistic mass equation I mentioned in posts 28 and 36. I am still not clear about the reasons why the equation is out of favour. I see it as being a useful equation in that amongst other things it can be used to calculate the KE of a body.
Can it be used to calculate KE? Does it give the right answers? If so apart from the out of favour terminology used, such as relativistic mass, what's wrong with the equation?

 

[atyy]

It's me again. I refer once more to the relativistic mass equation I mentioned in posts 28 and 36. I am still not clear about the reasons why the equation is out of favour. I see it as being a useful equation in that amongst other things it can be used to calculate the KE of a body.
Can it be used to calculate KE? Does it give the right answers? If so apart from the out of favour terminology used, such as relativistic mass, what's wrong with the equation?

The relativistic mass is not out of favour.

In many treatments, it is confusing to refer to relativistic mass and invariant mass, so the tendency nowadays is to call the former the energy and the latter the mass. However, both usages are useful to know, since one encounters it in introductory treatments like those of Einstein, French, Feynman, Purcell, Rindler, Schutz. Even MTW use the term "mass-energy". (Side note: Often the relativistic mass, which is the same as energy, is identified with the inertial mass. However, photons do not have inertial mass, but they do have energy or relativistic mass.)

Knowing both terms is still necessary in the advanced literature. For example, http://arxiv.org/abs/1001.5429 remarks "Remark 5. In the literature, references are found where the term ADM mass actually refers to this length of the ADM 4-momentum and other references where it refers to its time component, that we have named here as the ADM energy. These differences somehow reflect traditional usages in Special Relativity where the term mass is sometimes reserved to refer to the Poincare invariant (rest-mass) quantity, and in other occasions is used to denote the boost-dependent time component of the energy-momentum."

 

[Dadface]

Thank you very much to all who responded to my particular enquiry. Its good to see some backing for the equation I referred to. My knowledge and understanding of SR is very basic but I have always felt at home with the equation. I think I understand it, it feels right and it seems to work.

 

[Naty1]

I have two texts from Peter Bergmann. No bias against 'relativistic mass' from this student of Einstein!

In both texts, THE RIDDLE OF GRAVITATION [1992], and INTRODUCTION TO RELATIVITY [1976] he discusses definitions and differences between rest mass and relativistic mass without favoring one over the other.

From the first book as an example:

He says: [pg 40]

The mass of a body measured in a Lorentz frame in which it is a rest is called it's rest mass. The rest mass is an intrinsic property of a physical body, whereas the mass that governs its behavior in interactions with other bodies, it's relativistic mass, depends on the relative motion of object and observer as well. The sum of the relativistic masses of several interacting bodies remains unchanged through the interactions, but the rest mass can change.

He then explains as an example the radioactive decay of an atomic nucleus including the relativistic mass of an emitted gamma ray.

This thread leaves a very different impression from several others in these forums regarding 'relativistic mass'. Good to see varied viewpoints.

 

[Bill_K]

I have two texts from Peter Bergmann. No bias against 'relativistic mass' with this student of Einstein! In both texts, THE RIDDLE OF GRAVITATION [1992], and INTRODUCTION TO RELATIVITY [1976] he discusses definitions and differences between rest mass and relativistic mass without favoring one over the other.

The 1976 Dover edition of Peter's book is a reprint of what was the very first textbook on relativity (1942), and it still comes across as a very clear exposition. Plus, there's definite historical interest in what topics were emphasized at that time and how they were presented.

 

[harrylin]

There are some things that concern me about concepts of mass as discussed in threads such as this and I would appreciate it if someone could answer the following :

1. I have the impression that the majority of physicists who use relativity do not favour the equation:

M=MoL (L= Lorentz factor)

. I am a bit familiar with the other equations but is it considered that the above equation is archaic or misleading or incorrect in some way?

. If people reject the equation is it because of the terminology sometimes used? It seems to me that it is accepted that Mo can be referred to as the mass,or invariant mass and sometimes rest mass and that it is unnecessary to use the subscript o.
What doesn't seem to be accepted is that M (or E as it is sometimes written) should be referred to as the total mass where the total mass is the sum of the invariant mass plus the mass equivelent of the kinetic energy .If it is not accepted then what is wrong in calling it total mass and what, if anything, should it be called instead?

. Are there physicists who favour the use of the equation and if so are there examples of where the equation is more useful than any alternatives?
[..]

Hi Dadface,
the Physics FAQ gives a rather good discussion that I think answers all your above questions:

http://math.ucr.edu/home/baez/physic...y/SR/mass.html