Mass increasing as velocity approaches c

[Routaran]

I read that an objects mass increases as it approaches the speed of light and that this was the reason why anything that has mass can never travel at the speed of light. The energy required to accelerate it for that last step would become infinite.

my question is that if the mass of an object increases, wouldn't it collapse into a black hole before we had to worry about requiring infinite energy to accelerate it?
clearly i am wrong because that's not it so what am i missing?

 

[uby]

Prefacing this by noting that I claim no expertise on the subject:

By inputting energy in getting the particle to move up to some critical fraction of the speed of light, at some point you might not tell the difference in an inertial frame between a sufficiently large collection of particles with finite rest mass that can locally collapse spacetime on itself vs. a particle with large relativistic mass that can do the same. The only difference between the two cases would be the role that quantum uncertainty would play, which plays a big role in whether a black hole can form.

 

[jtbell]

FAQ: If you go too fast, do you become a black hole?

 

[danR]

A simpler answer than the FAQ item is that 'real' mass does not, in fact, increase, and PF has increasingly discountenanced that idea. The total energy of the system increases, and e=mc^2 declares an equivalency relationship between energy and mass, but not an equality.

A dollar bill is equivalent to a dollar in gold, but the bill is not gold.

 

[Routaran]

I see. So the initial statement that mass increases is not accurate. Where can I get more information on how gravity couples with momentum? I would like to learn more about what's really going on.

 

[elfmotat]

I see. So the initial statement that mass increases is not accurate. Where can I get more information on how gravity couples with momentum? I would like to learn more about what's really going on.

Here's the simple answer:

The equation:
$$E=\gamma mc^2$$
describes the relationship between the total energy of an object, its speed, and its invariant (rest) mass, where $\gamma=1/\sqrt{1-v^2/c^2}$.

Similarly:
$$\mathbf{p}=\gamma m\boldsymbol{v}$$
gives the momentum of an object.

Back when Special Relativity was new, physicists (I'm not sure whether or not Einstein came up with the idea) invented something called the "relativistic mass" which is given by $m_r=\gamma m$ so that the energy and momentum equations simplify to:

$$E=m_rc^2$$
$$\mathbf{p}=m_r \boldsymbol{v}$$

The relativistic mass increases with increasing velocity. The invariant (some would say "real") mass does not. Almost all physicists have abandoned the idea of relativistic mass nowadays because it has proven to be pretty much useless. The only people who haven't done away with it are authors of science books directed at people with little to no physics background because they think it makes things more intuitive. In my opinion, it only serves to confuse people who want to go deeper into the subject.

 

[DrStupid]

Back when Special Relativity was new, physicists (I'm not sure whether or not Einstein came up with the idea) invented something called the "relativistic mass" which is given by $m_r=\gamma m$ so that the energy and momentum equations simplify to:

$$E=m_rc^2$$
$$\mathbf{p}=m_r \boldsymbol{v}$$

$m_r$ was invented by Newton (see definition II) but it was invariant in classical mechanics and it was simply called mass. When Einstein replaced Galilei transformation by Lorentz transformation $m_r$ became velocity dependent.

 

[D H]

Emphasis mine:

The relativistic mass increases with increasing velocity. The invariant (some would say "real") mass does not. Almost all physicists have abandoned the idea of relativistic mass nowadays because it has proven to be pretty much useless. The only people who haven't done away with it are authors of science books directed at people with little to no physics background because they think it makes things more intuitive. In my opinion, it only serves to confuse people who want to go deeper into the subject.

Exactly. The question raised in the original post is proof of just that. Routaran, relativistic mass doesn't gravitate. It is better to think of invariant mass as the cause.

As far as why "almost all physicists have abandoned the idea of relativistic mass nowadays": Look at the equation for relativistic mass: $E=m_rc^2$. Relativistic mass is just a synonym for energy. Energy is a useful concept; relativistic mass, much less so.

$m_r$ was invented by Newton (see definition II) but it was invariant in classical mechanics and it was simply called mass.

That is too much historical revisionism. Newton knew nothing about special relativity; it was a couple of hundred years after his time. To say that Newton invented the concept of relativistic mass is worse than wrong; it is wronger than wrong. Newton didn't invent the concept of mass, either. Newton himself attributes his first two laws and his first several definitions to his predecessors.

 

[DrStupid]

Newton knew nothing about special relativity; it was a couple of hundred years after his time.

<irony>Really?</irony>

To say that Newton invented the concept of relativistic mass is worse than wrong; it is wronger than wrong.

Please carefully read what I wrote. I never claimed that Newton invented the concept of relativistic mass. In fact I mentioned that the mass as used Newton's definition II was invariant in classical mechanic and became relativistic in SR.