Infinite Mass, Infinite Energy

[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 traveling 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?

 

[jtbell]

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.

The first statement 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.

 

[WhyIsItSo]

The first statement doesn't say anything about relativistic mass. <snip>

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.

 

[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.

 

[pervect]

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 traveling 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?

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

 

[jtbell]

Why not?

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.

 

[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.

 

[JesseM]

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.

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.

 

[pervect]

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.

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.

 

[neutrino]

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.

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?

 

[WhyIsItSo]

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

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.

 

[selfAdjoint]

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?

True.

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

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.

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

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.

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

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.

 

[pervect]

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?

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 anyone 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.

 

[neutrino]

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.

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?

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.

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

 

[gijeqkeij]

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 traveling 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?

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

 

[pervect]

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

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.

 

[WhyIsItSo]

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.

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!

 

[WhyIsItSo]

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.

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

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.

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).

 

[jtbell]

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

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

2. Yes.

 

[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.

 

[WhyIsItSo]

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.

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?

 

[WhyIsItSo]

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

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?

 

[jtbell]

Different inertial frames could measure different velocities for the muon, right?

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&#039; + v

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

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

u = \frac{u&#039; + v}{1 + \frac{u&#039;v}{c^2}}

 

[WhyIsItSo]

u = u&#039; + v

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

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

u = \frac{u&#039; + v}{1 + \frac{u&#039;v}{c^2}}

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&#039; + 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?

 

[selfAdjoint]

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 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?

Yes if you know the muon's relative speed and have measured its momentum in your frame (hence its four-momentum), you can invert the Lorentz Transformation (they are all linear) to get its four-momentum in its rest frame, and that will be its rest mass, aka its invariant mass.



The English equivalent of u - u' + v is "u is the sum of u-prime and v". The relativity addition formula is "u is the quotient of the sum of u-prime and v, divided by 1 plus u times u-prime over c."

 

[WhyIsItSo]

Yes if you know the muon's relative speed and have measured its momentum in your frame (hence its four-momentum), you can invert the Lorentz Transformation (they are all linear) to get its four-momentum in its rest frame, and that will be its rest mass, aka its invariant mass.



The English equivalent of u - u' + v is "u is the sum of u-prime and v". The relativity addition formula is "u is the quotient of the sum of u-prime and v, divided by 1 plus u times u-prime over c."

Thank you kindly.

I see the phrase "Four-something" a great deal since I started learning about relativity. 4-momentum, 4-vector, and maybe others.

Is this three dimensional Newtonian space plus something else? Time?


u'... I've seen similar terminology in many math equations (derivitives I think). This symbology seems to have a different meaning. In current context, it seems to equate roughly to "a different-" or "some other-", like when I was reading yesterday one of Eisntein's addresses, he was talking about K inertial frame, and K' inertial frame.

When I see things like f, f', f''... This has a different meaning somehow, does it not?

 

[selfAdjoint]

Thank you kindly.

I see the phrase "Four-something" a great deal since I started learning about relativity. 4-momentum, 4-vector, and maybe others
Is this three dimensional Newtonian space plus something else? Time?.

Yes, a four-vector has four components, usually numbered 0, 1, 2, 3. The "0 component" is associated with the time coordinate, and the other thress with the three spatial coordinates. In fact the three spatial components are often split out as a three-vector (in which case you will see the bold letter notation for the vector, as is usual in ordinary vector analysis of three dimensional space; thus: four-vector u = (u^0,u^1,u^2,u^3) (note that the upper numbers are indices of components, not powers or exponents). Corresponding three vector \mathbf{v} = (u^1, u^2, u^3).

Each inertial frame will experience a four-vector with differing numerical components but the magnitude of the four-vector will be the same for every frame, or invariant as the term is. This magnitude is obtained by summing the squares of the components, after first multiplying the 0-component by c.

The splitting off of the three-vector is frame dependent (i.e. different for differently moving observers), as is ANY splitting of time and space apart in relativity.

Two well-known four-vectors are:
- The position four-vector (t, x, y, z) specifying the spacetime coordinates of some event, such as "12:00 Noon, Latitude 45 degrees North, Longitude 90 degrees West, Alltitude 300 feet."
- The four-momentum (E, p^1, p^2, p^3). The 0-component is energy, and the three spatial components are the components of the three-dimensional momentum. It is notable that the invariant magnitude of this four-vector (same in every frame, remember) is mc^2. And in the rest frame, the three-momentum is zero, so what do you get?

u'... I've seen similar terminology in many math equations (derivitives I think). This symbology seems to have a different meaning. In current context, it seems to equate roughly to "a different-" or "some other-", like when I was reading yesterday one of Eisntein's addresses, he was talking about K inertial frame, and K' inertial frame.

When I see things like f, f', f''... This has a different meaning somehow, does it not?

Yes, you are right. The use of the prime notation does have these two meanings, "derivative" and "some other" . Almost always the meaning is clear from the context, but if you have any doubts, this is a great place to ask for a clarification.

 

[WhyIsItSo]

The four-momentum (E, p^1, p^2, p^3). The 0-component is energy, and the three spatial components are the components of the three-dimensional momentum. It is notable that the invariant magnitude of this four-vector (same in every frame, remember) is mc^2. And in the rest frame, the three-momentum is zero, so what do you get?

I have read and re-read your question. Honestly, only the barest glimmer of understanding is dawning in my mind. I'm not sure what you are asking me.

Casting about myself for a possible answer, I suppose:
1. If momentum is 0 then this inertial frame considers itself at rest.
2. I suppose we could find the rest mass (invariant mass).
3. Time would be considered present time.
4. The Buccaneers won't make it to the SuperBowl this year.

I threw in (4) to indicate I really don't know what you are asking me. I assume there is some obvious point, but it is a mystery to me until I know more.

 

[pervect]

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?

True, though it would take a little more effort to clarify "constant output". Also note that with a rocket you'll be losing mass (by whichever defintion) due to expenditure of fuel. You could say that you wouldn't measure any change in mass other than the loss of mass due to fuel usage. You might replace the rocket with something that doesn't use fuel, like a light-sail to avoid this issue of loss-of mass due to the burning of fuel.

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

False. As we have explained, the invariant mass of the rocket would not change. The relativistic mass would change. Therfore you need to go into much more detail about how you are measuring the mass - just what are you measuring? If you want to be concise, you can just tell us which sort of mass you are measuring. If you are seeking physical insight, you need to describe the experimental arrangement you are using to measure mass, and we could tell you what sort of mass you were measuring. Note that depending on the experimental arrangement, you might be measuring something that is neither relativistic mass nor invariant mass.

 

[WhyIsItSo]

pervect,

I think you are trying to confuse :) It is working :(

To progress towards the real concept I want help with, assume I am talking about relatavistic mass; especially since, as I specified, you are taking measurements from Earth, in your inertial frame.

But then this doesn't fit, does it? Since I'm accelerating I am not in an inertial frame, therefore we must use general, and not special relativity, correct?

Does that change how I should be asking this question?

 

[WhyIsItSo]

There was an important question I asked that got overlooked.

With the increase of mass due to velocity, what is happening to density? Is the volume of the object increasing also?

 

[jtbell]

Is the volume of the object increasing also?

Remember length contraction? It applies in the direction parallel to the direction of motion, but the directions perpendicular to the motion are unaffected. So the volume of the object decreases.

 

[WhyIsItSo]

Remember length contraction? It applies in the direction parallel to the direction of motion, but the directions perpendicular to the motion are unaffected. So the volume of the object decreases.

Huh...thinking...thinking...

mass increasing + volume decreasing = density increasing?

Or is this one of those points were relativity surprises me once again?