Inertia and relativistic mass.

[JakeStan]

I've seen inertia described in a few ways and it is simply a mass' resistance to acceleration. But inertia in my mind starts to break down in my mind in relativistic terms because of mass-energy equivalence and the speed of light. As mass and inertia are directly related then shouldn't and object's inertial mass increase along with its relativistic mass at relativistic speeds?

Let's say I throw a baseball at 0.9c. Its mass will have increased due to its speed and thus so will its inertia. I'm fine until this point, but there are some questions that pop up. Wouldn't this imply that inertial mass can have a vector component? In the case I gave if the ball was flying at 0.9c along an axis and then I tried to move it orthogonally along that access (along an axis where its velocity was 0) then it would have less inertia than if i tried to accelerate it parallel to it's direction of movement.

Next, in the last statement the ball's velocity is relative to some observer. Since relativistic mass is relative to an observer, is inertial mass as well? If so then the amount of force required to change the velocity of an object moving in my inertial frame would depend on the magnitude (and direction as outlined in the last paragraph) of its velocity in that frame. If I were to try to accelerate the ball from a distance, the amount of force required would depend on its velocity.

My last question is in regards to relativistic mass and gravity. Again, as velocity increases so too does relativistic mass. What effect does this have on gravity? Would a baseball flying by me at 0.9c pull me towards it with a greater force than one flying by at 10km/h? On the same note, would the ball flying at 0.9c pull me towards it as it has more inertia while the ball flying at 10km/h be pulled towards me as I have more inertia? Does my relative motion to the ball have an effect on its gravitational pull?

I guess the biggest question here is does inertia have any other equations or theories besides f=ma? Very little seems to be known about it and the few instances I've seen don't go any further into the question than "it is a property of mass" or, as stated earlier, "it is a mass' resistance to acceleration".

Any insight would be helpful in this regard.

 

[Dale]

I guess you need to decide what you think the word "mass" refers to. Do you think it is an intrinsic characteristic of an object, like charge, or do you think it is a relationship between two objects, like distance?

 

[jtbell]

I encourage you to do a search on this forum for the term "relativistic mass". You will find lots and lots of discussion about it and other kinds of "mass" in relativity, and lots of disagreement about which kind of "mass" is the "best" kind, or most worthy of being called plain old "mass."

The problem is that in relativistic physics, there is no single quantity that "does" all the things that the classical non-relativistic mass can be used for. Even with F = ma, which most introductory physics textbooks use as a definition of mass, it makes a difference whether the force is parallel or perpendicular to the direction of motion, or at some other angle.

On forums like this one, different people with different backgrounds make different choices about which uses of the classical mass are most important and should be carried over into relativity as the definition of "the" mass. Physicists who actually work with relativistic particles (nuclear and high-energy particle physicists, mostly) have pretty much settled on the "invariant mass" which is often called "rest mass" in introductory treatments. Ask a particle physicist, what is the mass of that electron flying by at 0.99999c, and he'll almost certainly answer "511 keV/c^2" (i.e. the "rest mass"). I was in particle physics as a grad student, and I'm pretty sure I never heard any of my colleagues refer to "relativistic mass."

Still, this is only a convention. People who really understand relativity, regardless of which kind of "mass" they prefer to talk about, agree on the results of calculations of physically observable quantities (e.g. how much does this electron's path bend in a magnetic field of a specified strength?), although they may use different equations corresponding to the kind of "mass" they use.

 

[ZikZak]

Answering your questions in order:

1. Yes, if you insist that F must always equal Ma, then yes, M becomes a vector, because the coordinate accleeration for a given force will depend on direction. In particular, the "tangential mass" will be $\gamma m$ and the "longitudinal mass" will be $\gamma^3 m$. This is a good reason to never use "relativistic mass," because it is always introduced as being both $\gamma m$ AND being the reason you can't accelerate something (in the longitudinal direction, obviously) to c, and this is at best misleading and at worst, simply wrong.

2. It depends on what you would like to call "inertial mass." If you want to call it this weird vector, then I suppose it is relative, but I think you're ripping the concept of mass a new one if you do. Why are we insisting on F=ma anyway? Even in Newtonian physics it is not correct. The correct equation, even in Newtonland is $F=dp/dt$. This equation is still true in relativity, when p is defined as $p=\gamma m v$. So I vote we stick with that.

Note on 1 and 2: The reason the coordinate acceleration changes with increasing speed is not that the mass increases (it doersn't); it's that you're using a reference frame that is relativistic relative to the rest frame of the particle. Near c, frames are "denser." The same force causes less change of velocity... for kinematic reasons, that because of mass increase. If you were to measure it acceration from an inertial frame initially moving with the particle, you can use F=ma to your heart's content.

3. Gravity is generated by energy and momentum: their density and flux. It has little to do with mass (which is only one kind of energy, it turns out) except that most of the energy in an everyday body is in its mass. Relativistically, the gravity has little to do with mass, relativistic or otherwise, except that it is one factor in the energy and momentum that goes into the stress energy tensor.

The concept of "Relativistic Mass" is a terrible idea because, as you have discovered, it is misleading and wrong, because it is not the inertial mass (even though it is taught in high schools as the reason you cannot achieve c) and it is not the gravitational mass. It is, in fact, the total energy. So why not just call it the total energy and leave mass alone, so that it remains an invariant quantity. There is no such thing as "Relativistic Mass:" there is only (invariant rest) mass and total energy.

 

[JakeStan]

Thanks for the answers.

Zak, your answer to #3. In the example I gave then, a baseball flying by me would have a greater pull than one floating beside me? A moving ball has more energy than one at rest and thus more gravity. But that would also mean that gravity is relativistic (whether by relativistic mass or simply by total energy). Wouldn't that also depend where you are in relation to the direction the ball is flying in? (i.e. I would be pulled toward a ball flying towards me more than one flying by me).

I'm still unsure of the effect speed of an object would have on gravity, specifically the orientation of you versus the object versus the direction of the velocity. Even if you substitute relativistic mass with total energy which is the more accurate term, you will still get differing values for gravity dependent on relative velocities.

 

[Dale]

This is a really common misunderstanding, http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html" . Mass is essentially only one of the 10 independent terms in the stress-energy tensor. There are 9 others, which include terms for moving mass, but the point is that you cannot think of gravity being simply related to the mass.

You didn't answer my question above: do you think of mass as an intrinsic property of an object like charge or as a relationship between two objects like distance? The reason that I ask is that there are two common definitions of mass that are used in SR, one is a frame-invariant quantity that all observers agree on and therefore represents a property of the object, and the other is frame-variant and therefore represents a relationship between the object and the observer rather than a property of the object. You can pick either definition, but the scientific community prefers the invariant mass.

 

[atyy]

You may find these discussions helpful:

Rindler, Essential Relativity, p77 "Relativistic Inertial Mass" http://books.google.com/books?id=0J_dwCmQThgC&printsec=frontcover#PPA77,M1

Carlip, Kinetic Energy and the Equivalence Principle, http://arxiv.org/abs/gr-qc/9909014