Has relativistic mass gone out of fashion?

[Jonathan Scott]

jarsta: You've convinced me that "relativistic mass" is not like rest mass, due to the lack of symmetry in the former. Is there a modern physical definition of mass then? E.g. the inertia of a body in its rest frame?

This also begs the question: how is force defined?

This is all mostly a matter of terminology.

Instead of "relativistic mass", scientists now refer to the total "energy", of a particle, which is essentially the same thing expressed in energy units. They then use the word "mass" to mean what was previously called "rest mass". However, one could equally well refer to "rest energy" and this term is used as well.

Force is still the rate of change of momentum. In relativity, momentum is effectively energy times velocity (divided by c^2 if you want it in the usual units). If the particle has rest mass, this is also equal to the mass times the proper velocity. Note that a force with a component in the direction of motion will increase or decrease the kinetic energy as well as changing the velocity.

 

[Dadface]

Is't it so that the relativistic mass equation is still useful in that it gives a quick method of calculating relativistic KE?

 

[ghwellsjr]

Now I began to really wonder, can the concept of relativistic mass really be wrong or out of fashion. I am not an expert on relativity, but how is the ultimate speed limit, the speed of light, now explained if not with velocity dependent mass?
If the velocity dependent mass does not prevent a massive object of achieving the speed of light, what is going to do it?

I don't think you need mass or energy to explain why a massive object cannot achieve the speed of light. I think all you need is to understand how we establish the speed of an object. It is done by determining the distance the object has moved away from (or towards) an observer and applying that distance at a certain time. Distance is now defined as how far light travels in a particular period of time. For convenience, we use units where the speed of light is 1 such as minutes and light minutes or just the arbitrary ticking of a stable clock, ticks for time and light ticks for distance. The ticks are shown as dots. We also believe in the Principle of Relativity, Einstein's first postulate, that two inertial observers measure the speed of each other identically.

We can illustrate this on a spacetime diagram where the two observers are depicted as thick lines as they move away from each other at a constant speed. In the first case, the blue observer sends a radar or light signal to the red observer which reflects back to him. The blue observer sends the signal 1 tick after they separate and receives the echo at the fourth tick. He then takes the time between sending and receiving the signal, in this case, 4-1=3 and divides that by two which is 1.5 light ticks and applies that distance to the average or midpoint of his two measurements, in this case (4+1)/2 = 2.5 ticks. Now he can determine the speed as the distance divided by the time or 1.5/2.5 = 0.6c:

Applying the measurement of distance to the average of the sending and receiving signals is nothing more than Einstein's second postulate.

In the next spacetime diagram, we see how the red observer makes the same measurement of the blue observer as blue did of red, even though the red observer may have a different kind of clock that may tick at a different rate:

Now let's throw in a third black observer who is traveling in the opposite direction away from the blue observer but at the same speed:

It should be no surprise that the blue observer gets the same answer for the black observer as he did for the red observer because the only difference is the direction of travel.

And it should be no surprise that the black observer gets the same answer in measuring blue's speed as red did when he measured blue's speed, even though black's clock is ticking much faster than red's clock:

But where the surprise may come in is when black measures the speed of red as shown here:

Black sends his radar signal out at his clock time of 1 tick and waits until 16 ticks to get the echo. His calculation of speed is (16-1)/(16+1) = 15/17 = 0.882c. (You may have noticed that I left off 2 divisions by 2 as they just cancel each other out.) You may have thought that the answer should be 1.2c but it turns out it is less than c. As a matter of fact, we can add together any number of observers at any speed short of c and the final speed will remain under c.

Now what if an observer tried to measure the speed of an object departing from him at the speed of light (if such a thing were possible). Clearly, he could not use the same method as before since his outgoing radar signal could never catch up to the object as shown in this diagram:

 

[Jonathan Scott]

Is't it so that the relativistic mass equation is still useful in that it gives a quick method of calculating relativistic KE?

The equation is still important, but we now say that it gives the total energy for an item with a given (rest) mass and speed. You can therefore also calculate the kinetic energy by subtracting the rest energy.

Personally, I'm happy with using "relativistic" or "rest" in combination with "mass", and similarly using "total", "rest" and "kinetic" in combination with "energy". However, the important thing to note is that if we say "mass" on its own, the modern convention is that this means what used to be called the "rest mass", and if we say "energy" on its own in a relativity context, this means the total energy.

I'm not entirely happy with the way this modern terminology ignores factors of $c$, as although this simplifies SR I think this can add confusion to GR later.

In special relativity, energy and mass only differ by factors of $c$, which can be set to 1, so they are essentially equivalent. However, there is a complication in general relativity when describing events within a coordinate system, in that the coordinate speed of light can vary with direction. This means that coordinate values with dimensions of mass and energy actually vary in different ways with potential (and mass can be different in different directions).

When factors of $c$ are included explicitly and are treated as referring to the variable coordinate-dependent speed of light (which can of course be set to 1 in a local SR frame), various Newtonian equations of motion (relating for example to conservation of momentum, angular momentum and rate of change of momentum in free fall) have natural very similar and easily understood forms in GR.

For example, the rate of change of momentum of something with energy $E$ falling in a weak Newtonian gravitational field $g$ as represented in isotropic coordinates (where the scale factor between local space and coordinate space is the same in all directions) is given in GR by the following equation where all values are expressed as coordinate values including the coordinate speed of light $c$:
$$\frac{d \mathbf{p}}{dt} = \frac{E}{c^2} \mathbf{g} \left ( 1 + \frac{v^2}{c^2} \right ) $$
This equation holds regardless of the direction of $v$ (radial, tangential or somewhere in between), just like Newtonian gravity, with the only difference being the additional term due to the curvature of space. Even that term only depends on the speed, not the direction. (For tangential motion, it leads to a curved path, and for radial motion it causes a change in momentum due to the change of the coordinate scale factor of space).

This notation is of course very unconventional, as $c$ normally represents the standard speed of light, so any equations using it for the coordinate speed of light either need very clear explanation of this unusual use or need some alternative notation, such as primed variables.

Of course this doesn't change the physics, but to write the above equation in the GR way without using the coordinate speed of light one would have to insert various scale factors from the metric which would obscure the relationship with the Newtonian view.

As I like to remain aware of these relationships between Newtonian and GR quantities, I personally prefer to continue to use "mass" for quantities with dimensions of mass and "energy" for quantities with dimension of energy, with appropriate qualifiers like "rest" when necessary to avoid ambiguity and explicit factors of $c$ when necessary to convert between them.

 

[7777777]

Relativistic mass appears in the formula:

E^2 = p^2 c^2 + m_{rest}^2 c^4 = m_{rel}^2 c^4
E = m_{rel}c^2


these formulas are taken from link


One way to avoid writing relativistic mass is to write instead:

E = m_{rel}c^2 = \gamma m_{rest}c^2

where
\gamma = \frac {1} {\sqrt{1 - v^2/c^2}}

The problem arises if v = c is inserted into the above equation. The value of
\gamma becomes infinity because of division by zero.
The problem of dividing by zero can be avoided if v > c. It will lead to imaginary
value of \gamma, imaginary value of energy, whatever that means. It appears
that speeds exceeding c are possible at least theoretically (mathematically) as also ChrisVer said.
I found useful info on link. Jesse Berezovsky writes about the problem
of how E becomes 0/0 if v=c and m_{0}=0. He suggests looking at the limit of E
as mass goes to zero and v goes to c.
I am suggesting looking at v>c, and avoiding the relativistic mass m_{rel} ,
using instead the invariant mass, the nonzero rest mass m_{rest} = m_{0}.

 

[ChrisVer]

Well, you can also have imaginary mass without having superluminal propagation...http://en.wikipedia.org/wiki/Tachyon#Fields_with_imaginary_mass

 

[stevendaryl]

There was a discussion in Physics Forums that showed that the concept of "relativistic mass" was of some theoretical interest in understanding how the group theory of the Poincare group relates to the group theory of the Galilei group in the nonrelativistic limit.

You actually need an 11th degree of freedom to define the Galilei
limit. The 11th parameter (in Galilei) is the "central charge"
associated with mass. To make Poincare' suitable for taking the limit
to Galilei (that is: to implement the correspondence principle with
respect to non-relativistic physics!), one needs to split the energy
generator E into two parameters -- kinetic energy H and "relativistic
mass" M. E does not have a Galilean limit. The mass shell condition (E/
c)^2 - P^2 = (mc)^2 (where P is the momentum and m the rest mass)
needs to be generalized to P^2 - 2MH + (1/c)^2 H^2 = constant. An
additional invariant emerges: M - (1/c)^2 H = constant.

So it's not 100% correct that nobody cares about relativistic mass anymore. Maybe 99%.

https://www.physicsforums.com/showthread.php?t=174039