Mass in Special Relativity: Misconception or Reality?

[HungryBunny]

Hi PhysicsForum,

I'm currently reading Spacetime Physics by Taylor and Wheeler and I can't wrap my head around the concept of mass in SR. In the textbook, mass is described as the magnitude of the momenergy 4-vector and is invariant no matter which inertial reference frame you choose.

So does (relativistic) mass of an object increase with velocity or does mass remain invariant (since it would be possible to pick a reference frame at the same speed as the object and thus it's mass would only be its rest-mass)? The latter seems like the correct concept, and if so, how is the impossibility of an object to ever attain the speed of light explained? Would it be correct to say, an object moving at the speed of light would theoretically have an infinite amount of energy (E = mc²/√(1-v²/c²), denominator goes to 0 as v → c)?

Thanks for the time to read such a seemingly elementary question!

 

[Ibix]

Most people don't use relativistic mass anymore, so you can assume that mass means invariant mass or rest mass (three names for the same thing) until proven otherwise by the text.

The impossibility of reaching lightspeed comes directly from Einstein's postulates. The speed of light is frame invariant. No matter how long you accelerate for, there is always an inertial frame in which you are momentarily at rest - and in that frame light is going at c. Rinse and repeat...

 

[jtbell]

Would it be correct to say, an object moving at the speed of light would theoretically have an infinite amount of energy (E = mc2/√(1-v2/c2), denominator goes to 0 as v → c)?

You have the right idea here. A nit-pickingly better way to say it would be that as you give an object more and more energy, its speed becomes asymptotically closer and closer to c; but it's impossible to give the object an infinite amount of energy, so its speed can never reach c.

 

[HungryBunny]

Most people don't use relativistic mass anymore, so you can assume that mass means invariant mass or rest mass (three names for the same thing) until proven otherwise by the text.

The impossibility of reaching lightspeed comes directly from Einstein's postulates. The speed of light is frame invariant. No matter how long you accelerate for, there is always an inertial frame in which you are momentarily at rest - and in that frame light is going at c. Rinse and repeat...

You have the right idea here. A nit-pickingly better way to say it would be that as you give an object more and more energy, its speed becomes asymptotically closer and closer to c; but it's impossible to give the object an infinite amount of energy, so its speed can never reach c.

Ah, these two explanations make a lot of sense. Thank you for them!

May I ask 2 more questions?

1. In an isolated system of particles, the conservation of momenergy is actually equivalent to both the conservation of energy and momentum separately? i.e In picturing the momenergy 4-vector as a right-angled triangle, the x-axis (momentum), y-axis (energy) and consequently the Lorentzian hypotenuse (mass) all remain of the same magnitude (or rather, the right-angled momenergy triangle for the system remains completely unchanged no matter what collisions/interactions occur within the system)?

2. The textbook mentions that the conversion from energy to mass, related by a conversion factor of c², is merely a historical accident of convention (just like how meters and miles are human-defined by convention) rather than a 'deep new principle'. I don't think I fully understand this statement. How did we define energy and mass such that c² became the conversion factor by convention? It sure seems like a wonderful property of nature that we could link two seemingly different physical entities by a factor of another property of nature, c, squared.

 

[Dale]

i.e In picturing the momenergy 4-vector as a right-angled triangle, the x-axis (momentum), y-axis (energy) and consequently the Lorentzian hypotenuse (mass) all remain of the same magnitude (or rather, the right-angled momenergy triangle for the system remains completely unchanged no matter what collisions/interactions occur within the system)?

Yes

How did we define energy and mass such that c2 became the conversion factor by convention?

By choosing units such that $c\ne 1$ we have adopted a convention where a conversion factor is needed.

 

[Mister T]

So does (relativistic) mass of an object increase with velocity or does mass remain invariant (since it would be possible to pick a reference frame at the same speed as the object and thus it's mass would only be its rest-mass)?

When people use the term mass they almost always mean what some call the rest mass. It's the magnitude of the energy-momentum 4-vector. Relativistic mass does indeed increase with velocity, but the use of relativistic mass has a checkered past, was almost never used in high energy physics research, and has fallen out of use in other areas such as introductory physics textbooks in the last few decades.

Once you no longer use relativistic mass you are left with only the ordinary mass, and there's no longer any need to call it the rest mass to distinguish it from relativistic mass. Some call it the invariant mass just to be clear that they're not talking about relativistic mass.

We call $m$ the mass, so that what people used to call the relativistic mass is $\gamma m$ where $$\gamma \equiv \frac{1}{\sqrt{1-(\frac{v}{c})^2}}.$$

The latter seems like the correct concept, and if so, how is the impossibility of an object to ever attain the speed of light explained? Would it be correct to say, an object moving at the speed of light would theoretically have an infinite amount of energy (E = mc²/√(1-v²/c²), denominator goes to 0 as v → c)?

Certainly it's valid to say that the total energy of a particle is $E=\gamma mc^2$ and so $E$ approaches infinity as $v$ approaches $c$. As others have pointed out, though, it follows directly from the postulates. Imagine chasing a beam of light and trying to catch it. No matter how fast you pursue the beam it always recedes from you at speed $c$, thus you can never reach a speed of $c$.

The textbook mentions that the conversion from energy to mass, related by a conversion factor of c², is merely a historical accident of convention (just like how meters and miles are human-defined by convention) rather than a 'deep new principle'. I don't think I fully understand this statement. How did we define energy and mass such that c² became the conversion factor by convention?

Think of it this way. The joule was introduced as the $\mathrm{SI}$ unit of energy and the kilogram as the $\mathrm{SI}$ unit of mass. Later, when it was realized that rest energy is equivalent to mass, it became possible to use the same unit to measure both. Hence there's a conversion factor of $c^2$ when converting between the two units. But if you do indeed measure them both in the same units then in those unit systems $c=1$ and you essentially remove that conversion factor.