Could I really be as massive as the entire universe?

[p.tryon]

I recall a physics teacher claiming that if an object was to reach the speed of light its mass would be the same as the universe, and therefore the speed of light is a cosmological speed limit. Is this true?

I was discussing this with a student and we came to the following conclusions; Take for example a body that has a 90kg mass (such as me!):
since K.E. = 1/2mv^2 and e = mc^2

(a) as the body undergoes some small acceleration (e.g. from 0m/s to 1m/s) its mass must increase because its (kinetic) energy is increasing. Granted this will only be very small KE = 0.5 x 90 x 1^2 = 45J this equates to an increase in mass of 45J/9E16 = 5.0E-16kg

(b) Now imagine the body accelerates from 1m/s to 2m/s. The previous increase in mass should be added to the 1kg mass if the K.E. equation was reapplied to work out the change in energy of this second acceleration (from 1m/s to 2m/s). The change in energy will therefore be slightly greater even though the velocity increases by the same amount (1m/s).

(c) We agreed that feeding a continually increasing mass back in the equation for kinetic energy would result in an even bigger increse in mass as the object accelerates further (perhaps an exponential increase?) Furthermore, a continually increasing v-squared term would also result in a greater increase in energy (therefore mass).

Is this resoning correct? Would these effects necessarily bring my mass to the same as the universe were I to reach speed of light? This would truly be a dieters nightmare!

 

[jtbell]

You've got the right idea, loosely speaking, although the details are off because the relativistic kinetic energy formula is not

$$K = \frac{1}{2} mv^2$$

but rather

$$K = \frac {m_0 c^2}{\sqrt {1 - v^2 / c^2}} - m_0 c^2$$

where $m_0$ is the "rest mass."

The "relativistic mass" of an object is

$$m_{rel} = \frac {m_0}{\sqrt{1 - v^2 / c^2}}$$

so we can also write the kinetic energy as

$$K = m_{rel} c^2 - m_0 c^2$$

Note that you need to beware when reading posts in this forum (and elsewhere) that when some people say "mass" they mean $m_{rel}$ and when other people say "mass" they mean $m_0$.

Also note that as v gets closer and closer to c, $m_{rel}$ becomes bigger and bigger without limit, and so does the kinetic energy. So, no matter how much kinetic energy you give an object (i.e. no matter now much work you do on it), its speed can never reach exactly c, but must always be less than c, even if by only a smidgen.

 

[Lyuokdea]

As you start moving close to the speed of light, normal Newton's laws equations for energy as a function of velocity no longer apply. We now use the equations of special relativity,

$$E=\gamma m_{0}c^2$$

is the relativistic correction to the total energy (or total particle mass), where m_0 is the mass of the particle at rest, and:

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

We can see, that as a particle with mass > 0 moves closer and closer to the speed of light (in our reference frame), the energy (and also the observed mass) of the particle approaches infinity.

So your basic premise is correct, as you continue to move closer and closer to the speed of light, your mass (from the perspective of observers in some stationary reference frame), approaches infinity. The numbers are just adjusted by the relativistic correction (which also contains the fact that your velocity cannot exceed light, because there would be a negative number under the square root)

Edit: I should apparently refresh pages before I start posting replies.

~Lyuokdea

 

[Dale]

Hi p.tryon,

The concept of "relativistic mass" which you are using is a redundant concept that is not in general useage among modern physicists. As you mentioned several times the relativistic mass is the same as the total energy (particularly in units where c=1), so we don't really need two names for the same concept. Instead, when most modern physicists refer to "mass" they mean the rest mass mentioned by both of the previous two posters.

So, you can restate your first sentence thus: "I recall a physics teacher claiming that if an object was to reach the speed of light its energy would be the same as the universe, and therefore the speed of light is a cosmological speed limit. Is this true?" Which is not quite true. Even if it had as much energy as the entire universe it would still be less than light speed.

 

[Andrew Buren]

As the student in question I would like to know what the equation for the whole encreass mass/energy thing. And what are the different parts of it?

 

[Dale]

Hi Andrew, Welcome to PF!

The general equation which relates energy, mass, and momentum in Special Relativity is:

E² = (pc)² + (mc²)²

where E is the total energy, p is the momentum, m is the rest mass (what jtbell and Lyuokdea wrote as m0), and c is the speed of light. This is the general formula for energy and it applies for both massive and massless particles, but for massive particles it simplifies to the equations given by Lyuokdea.

As far as what the different parts of the equation are, obviously on the left-hand side is the total energy, which is related on the right hand side to the kinetic energy (pc) and the rest energy (mc²). That last term, the rest energy, is what is meant by mass-energy equivalence, and you see that in the rest frame of a massive particle p=0 so the general equation reduces to the famous E=mc².

 

[Andrew Buren]

So from this you culd find out how much mass one kg as you incease the speed be one meter per second

 

[Dale]

The m in the equation I gave is the rest mass which is invariant. So if an object at rest has m = 1 kg then at 1 m/s it will have m = 1 kg and at .99 c it will have m = 1 kg also. Refer to my response in post #4 above. Relativistic mass is not used by mainstream physicists, so standard terminology is that mass does not increase as speed increases.

What does increase is energy. So using Luyokdea's simplified formula for massive objects we have that a 1 kg mass at rest has an energy of 89,875,517,873,681,764 J and a 1 kg mass moving at 1 m/s has an energy of 89,875,517,873,681,764.5 J and a 1 kg mass moving at .99 c has an energy of 637,110,654,110,437,000 J.