# Infinite Mass, Infinite Energy

1. Sep 11, 2006

### WhyIsItSo

I frequently read that accelerating a rest mass to c is impossible because it requires infinite energy. I just read another such post a moment ago.

Surely, this must be a perception ONLY! That something's mass increases, I mean. It seems to me its mass does not actually increase, only appears to from the perspective a a relatively at rest observer. That in fact it is tied up with time dilation.

To a photon, which is travelling at c, the universe must be perceived as having infinite mass.

We would argue that any given object has some specific, measurable mass.

So it is just perception, right? The mass is not actually increasing?

2. Sep 11, 2006

### Staff: Mentor

The first statment doesn't say anything about relativistic mass. It's a consequence of the work-energy theorem, which says that the amount of work needed to increase an object's speed equals the increase in the object's kinetic energy: $W = K_{final} - K_{initial}$, which is still valid relativistically; and the relativistic formula for kinetic energy:

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

These give

$$W = \frac{m_0 c^2}{\sqrt{1 - v_{final}^2/c^2}} - \frac{m_0 c^2}{\sqrt{1 - v_{initial}^2/c^2}}$$

Calculate the work needed to accelerate an object from 0 to 0.9c, then from 0 to 0.99c, then from 0 to 0.999c, etc. You'll see that it increases without limit as the final speed approaches c.

For another perspective, calculate the work needed to go from 0 to 0.9c, then from 0.9c to 0.99c, then from 0.99c to 0.999c, etc.

Last edited: Sep 11, 2006
3. Sep 11, 2006

### WhyIsItSo

Why not?

If I jump on a rocket ship and start zipping away from Earth, and you stay here and observe, you'd be confirming that equation you used. You would conclude my mass (and that of the rocket ship) was increasing.

From my perspective, nothing has changed. My mass and that of my "environment" is unchanged.

Your perspective and mine would not agree.

4. Sep 11, 2006

### MeJennifer

In special relativity any measurement in a local intertial frame is independent of its relative motion.
There will be no changes, local time runs as usual, local distances are as usual and local mass is as usual!
This is obvious since for a local inertial frame one cannot say how fast it is moving or how close to c it is moving. One can only say how fast it is moving relative to something that has mass.

Relativistic effects, such as time dilation, space contraction and (relativistic) mass increase, will only be observed in a frame that is in relative motion with the frame that makes the observation.

The reason by the way that nothing can accelerate to c is that the speed of any local inertial frame relative to the speed of light is always c. We can only increase our speed relative to something that has mass but that speed increase is hyperbolic not linear, in other words it approaches c but never becomes c.

Last edited: Sep 11, 2006
5. Sep 11, 2006

### pervect

Staff Emeritus
It's a matter of defintion. Objects do have a mass that is independent of the frame of observation. This mass is called the invariant mass, and it is defined so that it does not change with velocity.

The relativistic mass is defined in such a manner that it is NOT independent of the observer.

I tend to greatly prefer the concept of invariant mass, because it is observer independent.

See for example http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

also the wikipedia article discusses the same issues somewhat

http://en.wikipedia.org/wiki/Rest_mass

6. Sep 11, 2006

### Staff: Mentor

I worded my response poorly. I thought you were trying to argue against the idea that "accelerating a rest mass to c is impossible because it requires infinite energy," by bringing in the idea of relativistic mass somehow. I was trying to point out that the statement in quotes doesn't rely on the concept of relativistic mass.

The relativistic mass simply increases along with the object's energy. The energy and relativistic mass of an object both depend on the reference frame they're measured from. But no matter what inertial reference you're observing the object from, it can't be accelerated to a speed of c, because it would require an infinite amount of work in that frame.

If you do have a problem with "accelerating a rest mass to c is impossible because it requires infinite energy," then it's not clear to me what that problem is.

7. Sep 11, 2006

### WhyIsItSo

The key to my question is that the assertion that mass changes with velocity seems to me to be a perception only. No actual mass change occurs.

That is the concept I'd like to see clarification on.

8. Sep 11, 2006

### JesseM

In Newtonian physics, inertial mass can be defined in terms of the amount of force or energy to accelerate a mass by a given amount. Even if you choose not to use the concept of relativistic mass (and most physicists nowadays don't, in fact), it's still true that for an object with a given rest mass, it takes more and more energy to accelerate it by a fixed amount in its direction of motion as its speed gets closer and closer to c. You can derive this fact without making any reference to relativistic mass, using something like the equation $$E^2 = m^2 c^4 + p^2 c^2$$, where m is the rest mass and p is the relativistic momentum.

Last edited: Sep 11, 2006
9. Sep 11, 2006

### pervect

Staff Emeritus
There are two different defintions of mass. One is observer dependent, the other isnt. What do you think needs to be clarified?

Note that "reality" is a philosophical concept, not a physical concept. Debates about what is "real" and what is "not real" can't be resolved by experiment, and usually just lead to endless discussion.

10. Sep 12, 2006

### neutrino

Then, what is it that we find when doing experiments? Of course, we don't see atoms as tiny billiard balls, but what are we doing when we observe the results of an experiment? Is it not a "reflection" of reality?

11. Sep 12, 2006

### WhyIsItSo

Let me put it this way. "I" or "me" refers to myself accelerating away from Earth. "You" refers to any observer on Earth monitoring my progress.

1. As long as my fuel lasts, given a constant output, I'll measure a constant acceleration. I would not measure any change in my mass. True or false?

2. Assuming the means to monitor and take whatever measurements you required, you would note my mass increases. True or false?

The clarification. If, as I understand it, both statements are true, then one must be a perception and not fact.

I have further concepts I wish to develop, but I will wait to see if I am corrected on this point first.

12. Sep 12, 2006

Staff Emeritus
True.

The increase can be attributed to mass or to energy with a constant mass. Why don't you use length or time instead? These are not so controversial, and I think they gt to your point just as well.

Wrong. There is no finally definite value of the Lorentz transformed quantities. They are all established relative to their observers and they are real, in the sense that if you do an experiment that requires two different frames to interact, the transformed values will be the ones that determine what happens. Classic example; measuring properties of muons from cosmic rays. The muons are moving very fast relative to the measurement equipment and the Lorentz transformed lifetime is measured.

Rather than trying to develop your own physics in ignorance of the experimental and theoretical tradition, you would do better to study what has been found.

13. Sep 12, 2006

### pervect

Staff Emeritus
To phrase your problem so that it can be adressed scientifically, rather than philosophically, you need to specify the sorts of experiments that you are doing to "measure the mass" of an object. Without specifying the experiments, one cannot make any correct statements about the properties of the results. I.e. there is no particular reason to believe that an experiment "measuring mass" will give an answer independent of a velocity of a particle unless that experiment is measuring the sort of mass (invariant mass) that one expects to exhibit this property.

Perhaps this is what you are also seeking to have clarified, i.e. knowing that there are two defintions of mass aren't enough, you need to know how each sort is measured. (And perhaps not, it's hard to say for sure what your agenda is).

Unfortunately, there are a lot of different ways one might go about attempting to measure mass, and depending on the exact measurement setup one might wind up measuring any one of relativistic mass, invariant mass, or even longituidinal or transverse masses.

To measure the invariant mass of a system, one measures its energy, and its momentum, and applies the energy-momentum equation to compute

m = sqrt(E^2 - (pc)^2) / c^2

One will find that this quantity does not change, regardless of how the particle is moving.

Transverse mass for a particle will be numerically equal to its relativistic mass, and could be measured by applying a transverse force to the particle (perhaps a charged particle with a magnetic field) and measuring its transverse acceleration, i.e. the amount the particle was deflected. [add] Transverse mass will depend on the velocity of the particle.

If one uses forces in the direction of motion (longitudinal forces) rather than transverse forces, and measures force/acceleration, one gets the so-called longitudinal mass. [add] Longitudinal mass will also depend on the velocity of the particle. The velocity dependence will be different than that of transverse mass.

Other sorts of expriements are possible and interesting, for instance one might be interested in the velocity acquired by an object initially at rest if another massive object does a relativistic flyby. The answer in this case turns out to be "none of the above", though there is a formula for the predictions of GR in this case that I can dig up if anybody is interested.

Last edited: Sep 12, 2006
14. Sep 12, 2006

### neutrino

In a way, that more or less answers my question, although I did not confine it to this particular case (relativistic increase in mass). But the examples you have provided does shed some light on experiments in general. I guess this is somewhat similar to asking whether light is a wave or a particle - it depends on how you measure and what you want to measure, right?

I would be interested, but I'm afraid I'm not quite ready for GR yet.

15. Sep 12, 2006

### gijeqkeij

Few thoughts
* mass in GR is not actually defined and mass density is an invariant but not a conserved quantity
* in SR you can imagine a frame of ref. like a grid with no limit in the extension but in SR you can't consider mass
* in GR, where heavy objects are present, you can't consider a frame of ref. like the one of SR but only "local" frame of reference (so the statement of what the photon see of the universe doesn't make much sense)
* your last question should be on energy: the ENERGY is not actually increasing? and the answer is: yes it is

gijeqkeij

Universe it's so simple that is almost impossible for us to understand it

16. Sep 12, 2006

### pervect

Staff Emeritus
The full paper isn't available for free on the web anyway. The abstract is not terribly technical and is at the link below.

http://dx.doi.org/10.1119/1.14280

I'm not sure how well known and accepted the particular name the authors have chosen for this measurement is, i.e. I suspect that if you talked to a random physicist about "active gravitational mass" without referencing this paper you'd get a lot of blank looks. The abstract makes it reasonably clear what sort of measurement is being made, though.

17. Sep 12, 2006

### WhyIsItSo

If I've been reading correctly today, this is not a special, but a general relativity scenario. It seems to me that special relativity, as you say, is about inertial frames, while my frame is accelerating!

18. Sep 12, 2006

### WhyIsItSo

Two questions.
1. Are the muons travelling at less than c?
2. Does the Lorentz transformation describe the relationship between two different (inertial?) frames?

I'm not developing my own physics, I'm developing my concepts; two very different things. I suppose I do poke and prod at "accepted" truth, but then, why not? It is an effective way to really understand the subject (as opposed to learning it by rote).

19. Sep 12, 2006

### Staff: Mentor

1. Yes, no matter which inertial reference frame you measure them in.

2. Yes.

20. Sep 12, 2006

### waht

Mass is simply a tendency to resist motion.

If you apply 1 N to an object, it will accelerate let's say 10 m/s^2

so it's mass (or resistance to motion) is F/a = .1 Kg

At non-relativistic speeds, the ratio F/a is pretty linear, that's why mass appears to be contant.

But at speeds close to c, the resistance to motion will increase exponentially, you see this as mass a increase, or an energy increase.

21. Sep 13, 2006

### WhyIsItSo

That's an interesting definition of mass. I'll have to ponder that for a while.

It raised a question, however. If relativistic mass increases, what is going on with density? Is the volume also increasing?

22. Sep 13, 2006

### WhyIsItSo

As a reminder, your answer was in response to my question about the speed of the muons.

I assume this is due to the muon having mass, hence can never reach c.

By specifying this is true regardless of intertial frame, it has raised another question, well, clarification really. Different inertial frames could measure different velocities for the muon, right?

Let's see. Another question. If I've followed, then the mass of a muon has been experimentally measured, as has its velocity as measured from our intertial frame. Is that all the information required to use the Lorentz transformation to derive the muon's rest mass?

23. Sep 13, 2006

### Staff: Mentor

Yes. I suspect that you need to learn what inertial reference frames are, and what it means to measure an object's velocity relative to an inertial reference frame. These concepts apply to classical mechanics as well as to relativistic mechanics, so here's a concrete example from classical mechanics.

Imagine a straight, level road. Car A travels at a constant 50 km/hr, and car B travels at a constant 70 km/hr in the same direction. Both velocities are measured with respect to the ground by the cars' own speedometers. Both cars travel in a straight line at constant velocity, so we have two inertial reference frames, A and B, associated with the cars. The frames are "inertial" because the cars are not accelerating or decelerating or going around a curve. If we are riding in either car with our eyes closed, we don't feel any forces associated with acceleration, and in fact we can't tell whether we are moving or not, without looking outside the car, or at the speedometer.

Now imagine a third car, C, whizzing past the other two cars, going in the same direction at 80 km/hr with respect to the ground. From the point of view of car A, car C is traveling at 30 km/hr, so this is the velocity of car C in inertial reference frame A. Similarly, the velocity of car C in 10 km/hr in inertial reference frame B.

Finally, imagine a muon traveling at some velocity with respect to the ground. Just as with car C, it has different velocities in inertial reference frames A and B.

The discussion above would be the same if we were using relativistic mechanics instead of classical mechanics, except for one thing. In the discussion above, we simply "add" or "subtract" velocities to switch between inertial reference frames. Explicitly, if the velocity of car C relative to the ground is u, and the velocity of C relative to frame A is u', and the velocity of frame A relative to the ground is v, then:

$$u = u' + v$$

(in this example, 80 = 30 + 50.)

In relativity, this "velocity addition" proceeds differently. We have to use instead:

$$u = \frac{u' + v}{1 + \frac{u'v}{c^2}}$$

Last edited: Sep 13, 2006
24. Sep 13, 2006

### WhyIsItSo

That part I think I get. I have an appreciation of inertial frames. My request for clarification comes more from becoming paranoid about making "logical-seeming" assumptions. I've stubbed my toe badly a few times here by doing so, and been chastised accordingly :(

You did miss my last question; is knowing the mass and velocity of the muon according to our frame sufficient information to determine the muon's rest mass (and I take that to mean its mass in its inertial frame)?

And a new question somewhat diverging... Regarding the math, how would you express $$u = u' + v$$ in words? As in, if you were talking to me on the phone, what would you say to tell me that equation? Is that u' called "u prime" for example?

25. Sep 13, 2006