# If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?

I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.

As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?

If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?

I quote Igor Ivanov (a stackexchange user from this question Why does the (relativistic) mass of an object increase when its speed approaches that of light?) to give an insight into my question.

The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".

EDIT

After reading the answers I have a doubt.

SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy.

• There is not such a thing as 'observable mass'. The mass is the mass. In modern physics we don't speak about 'rest' mass or 'observable mass'. Oct 11 '14 at 10:04
• I know but I meant relativistic mass. Oct 11 '14 at 10:16
• In modern physics we don't make any distinction between any type of masses (in the relativistic context). We only use the mass of the particle (the $0.5\text{ MeV}$ of the electron, for example). The relativistic energy depends on this mass, and this mass doesn't change, but depends too on the velocity, in such a way that is not allowed to trespass $c$, and infinite energy is needed to reach $c$. This has nothing to do with any (non-existing) variation in the mass. Oct 11 '14 at 10:34

If rest mass does not change with v then why is infinite energy required to accelerate an object to the speed of light?

The momentum of a material particle, a conserved quantity, is theoretically and experimentally a non-linear function of velocity given by

$$\vec p = m \frac{\vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

which goes to infinity as $v \rightarrow c$. The relativistic energy, also a conserved quantity, is

$$E = c\sqrt{|p|^2 + (mc)^2}$$

When a particle is ultrarelativistic, $|p|^2 \gg (mc)^2$, this expression is approximately

$$E = |p|c$$

which has been experimentally confirmed. So, it easy to see that the particle's energy goes to infinity as $v \rightarrow c$

• SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, Then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy. Oct 12 '14 at 4:22
• @rahulgarg12342, the answer I gave clearly shows why the energy goes to infinity. If you're asking why these equations for momentum and energy correspond to experimental results, you're asking a metaphysics or philosophical question, not a physics question. Oct 12 '14 at 14:17
• Yes I was asking a metaphysical question. Can you please answer it? Oct 13 '14 at 9:40
• @rahulgarg12342, if you know that your question is metaphysical, you must know that I don't have an answer to it other than "because the Universe is what it is and not something else". Oct 13 '14 at 10:55
• So we just have to admit, That experimentally more and more energy is required? Oct 13 '14 at 11:16

In relativity the rest mass is the mass of an object measured from a reference frame in which it is at rest. But this is not the mass involved in acceleration or inertial mass.

Inertial mass, or the opposition of the body to the change of movement (directional or in magnitude), will grow with the speed of the body:

$$m = \frac{m_o}{\sqrt{1-v^2/c^2}}$$

and this is the mass observed when the object is moving. By observed I mean the way you would measure it, which could be: applying a force $F$ to it and measure the acceleration produced $a$ and you would get $m = F/a$.

Also, it is a mistake to think:

I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light

No body can travel faster than light, not because we can not state its speed, but because it can't. This is at least what our current state of knowledge leads to, with many experimental confirmations.