my point is:

How can we calculate invariant mass?? What is the still(inercial) state??

This question comes from several books and web sites that i've search about it, and my problem was; ok: with velocity, a body makes is m turn to M by Y factor, but, what can we assume is the m???? is the mass we calcule over volume and volumic mass? is the mass we height on earth using "g" aceleration??
because, as the earth itself is moving, then m is already being turned into M, and problaby, any other measurement is dependent of moving earth...

Another question about this is: considering E=Ymc^2 when v=/0, if a total aniquilation of a mass body hapens, what of this persons receive more E? A static person(considering referencial the mass body one), or a moving person?

My point of this question is: total energy depends on Y factor, so, if a mass body is still in relation to "A" and moving in relation to "B", "B" will receive more energy than "A" right? So in that case, the more faster we run from the explosion, the more energy we receive....:grumpy:

Once again, thanks...
Littlepig

Briefly - every mass or body has something called 'rest mass', m0. This is the mass of the thing in its own frame of reference. The 'rest energy' of the thing is m0*c^2. Generally energy is not measured the same between inertial frames, because a moving mass has kinetic energy as well as rest energy.
It's done much better here -

http://www2.slac.stanford.edu/vvc/theory/relativity.html

pervect
Staff Emeritus

There are at least two sorts of mass, which you appear to be confusing. Invariant mass is a property of the body itself, and does not change with speed. So called "relativistic mass" does change with speed (the gamma factor, i.e. $\gamma$. Because it changes with speed, it is not a property only of the body, but of the body and the observer.

Measuring the rest mass of an object is not as simple as it looks.

From the oustide an object could appear to be at rest but inside it is not. There can be cases where electrons reach relativistic speeds. For instance in the case of gold.

Obviously the difference is usually extremely small, so we don't have to worry that if we order 1kg of apples that we are cheated on because they sold us the relativistic instead of the rest mass.

Briefly - every mass or body has something called 'rest mass', m0. This is the mass of the thing in its own frame of reference. The 'rest energy' of the thing is m0*c^2. Generally energy is not measured the same between inertial frames, because a moving mass has kinetic energy as well as rest energy.
It's done much better here -

http://www2.slac.stanford.edu/vvc/theory/relativity.html

Note: The term invariant mass is often meant to refer to the proper mass of a system of particles although many people use them to mean the same thing for the obvious reasons. Of course there is nothing wrong with this. I was een unaware of this until the author of one of my GR texts told me that's what he meant when he refered to invariant mass.

There was a time when this subject came up a lot. For that reason I created a web page to describe the topic. Not everything is in there because I wrote that several years ago and I've learned a lot about the topic so far.

Measurement is still something I've quite gotten yet. E.g. to measure invariant mass of a particle (e.g. the proper mass of a particle) then one might think that all you need is to measure the energy, E, and momentum, p of the body. Measuring the momentum doesn't seem troublesome to me but measuring the energy does see to be problem some since to determine the energy it appears that one needs to measure the rest mass, leading you into a circle. The measurements that are measued in particle accelerators is kinetic energy. IT appears that the the proper mass can be determined from the kinetic energy, hence no logical circle.

Anyone have thoughts on this? Thanks.

Pete

jtbell
Mentor
How can we calculate invariant mass?? What is the still(inercial) state??

In any inertial reference frame you like, measure the object's momentum p and energy E. Its invariant mass is then

$$m_0 = \frac {\sqrt {E^2 - (pc)^2}} {c^2}$$

It doesn't matter which inertial reference frame you use. In different i.r.f's the energy and momentum have different values, but $m_0$ as calculated above always comes out to the same value. That's why we call it the "invariant mass."

In any inertial reference frame you like, measure the object's momentum p and energy E. Its invariant mass is then

$$m_0 = \frac {\sqrt {E^2 - (pc)^2}} {c^2}$$

It doesn't matter which inertial reference frame you use. In different i.r.f's the energy and momentum have different values, but $m_0$ as calculated above always comes out to the same value. That's why we call it the "invariant mass."
How do you measure E without needing to know m0?

Pete

pervect
Staff Emeritus
How do you measure E without needing to know m0?

Pete

An experimental way to do it would be to do a curve fit. You can measure relative energies - so you just plot a curve of E_relative vs p, and look for the value of E_0 that makes E^2 - p^2 (in geometric units) constant.

mass relativity

There are at least two sorts of mass, which you appear to be confusing. Invariant mass is a property of the body itself, and does not change with speed. So called "relativistic mass" does change with speed (the gamma factor, i.e. $\gamma$. Because it changes with speed, it is not a property only of the body, but of the body and the observer.
Do you think that using for invariant mass, Newtonian mass would be confusing?
use soft words and kind arguments

Measuring the rest mass of an object is not as simple as it looks.

From the oustide an object could appear to be at rest but inside it is not. There can be cases where electrons reach relativistic speeds. For instance in the case of gold.

Obviously the difference is usually extremely small, so we don't have to worry that if we order 1kg of apples that we are cheated on because they sold us the relativistic instead of the rest mass.

I try to understand your point of view. When gold is in a state of rest relative to me i measure its rest mass which includes the equivalent of the kinetic energies of the electrons moving inside it, Where from could the difference you mention arrise.
use soft words and hard arguments:rofl:

mass relativity

An experimental way to do it would be to do a curve fit. You can measure relative energies - so you just plot a curve of E_relative vs p, and look for the value of E_0 that makes E^2 - p^2 (in geometric units) constant.

The first place where I have met the ideea to fit experimental results is
Uri Haber-Schaim, 'The teaching of relativity in the senior high school."
The Physics Teacher February 1971 p.75
I think that it is worth to extend the ideea.

An experimental way to do it would be to do a curve fit. You can measure relative energies - so you just plot a curve of E_relative vs p, and look for the value of E_0 that makes E^2 - p^2 (in geometric units) constant.
I don't understand. What is "E_relative" and how do you do to measure it? If a concrete example is needed then use an electron as an example.

Best regards

Pete

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Yeahh, thats very nice...ahaha, and they said poetry were in languages curse...

however, M(relativistic mass) is what counts in inercia, reason why c is impossible to reach, as M tends to infinite...

another question is, in spacetime curvature(gravity expanation by general relativity), what counts? invariant or relativistic mass?
for instance, what curves more the space? a still apple or a moving one?
I read about matter curves spacetime, so can M(relativistic) be considered matter? or as it depends on observer, it isn't?

Another point, just to see if i'm right: measuring M and m of a 0.8c object, M=/m, however, if we manage to acelerate to 0.8c, as we aproach 0.8c, M tends to m...when we reach 0.8c, we actually measure that M=m. so basically we measure a decrease of object's relativistic mass has we acelerate...right?

thanks for all, and enjoy your weekend...

however, M(relativistic mass) is what counts in inercia, reason why c is impossible to reach, as M tends to infinite...
This is a very common misunderstanding. The reason you cannot reach c has nothing to do with relativistic mass going to infinite.
When you change relative speed you use the velocity addition formula. In this formula the velocity does not go to c in the limit, in fact it has no limit.

what curves more the space? a still apple or a moving one?
Motion is relative not absolute, so the question is meaningless.

I read about matter curves spacetime, so can M(relativistic) be considered matter? or as it depends on observer, it isn't?
Curvature does not depend on the observer, the curvature depends on the rest mass and energy.

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JesseM
Measuring the rest mass of an object is not as simple as it looks.

From the oustide an object could appear to be at rest but inside it is not. There can be cases where electrons reach relativistic speeds. For instance in the case of gold.
I think that for composite systems involving multiple particles like a gold atom, the "rest mass" is defined in terms of the system's resistance to acceleration in the frame of the object's center of mass (which is proportional to the system's total energy in that frame, including kinetic and potential energy as well as the rest mass energy of each particle)--so, for example, a heated brick is defined to have a greater "rest mass" than the same brick when cool, even though all the individual particles making it up have the same rest mass in both cases.

robphy
Homework Helper
Gold Member
This is a very common misunderstanding. The reason you cannot reach c has nothing to do with relativistic mass going to infinite.
When you change relative speed you use the velocity addition formula. In this formula the velocity does not go to c in the limit, in fact it has no limit.

It is better to say "In this formula, the [magnitude of the] velocity never reaches the upper-bound, c".

I think that for composite systems involving multiple particles like a gold atom, the "rest mass" is defined in terms of the system's resistance to acceleration in the frame of the object's center of mass (which is proportional to the system's total energy in that frame, including kinetic and potential energy as well as the rest mass energy of each particle)--so, for example, a heated brick is defined to have a greater "rest mass" than the same brick when cool, even though all the individual particles making it up have the same rest mass in both cases.
Well if it is the definition then there is nothing to argue I suppose.

Arguing definitions does not make anyone wiser.

It is better to say "In this formula, the [magnitude of the] velocity never reaches the upper-bound, c".
Thanks Robphy!

This is a very common misunderstanding. The reason you cannot reach c has nothing to do with relativistic mass going to infinite.

it's pretty hard to see something approaching infinite inertial mass to accelerate further. also $c^2$ times that relativistic mass (the mass as it appears to the "stationary" observer) is the total energy of the body as it appears to the same observer. putting in an infinite amount of energy to get the body to accelerate to the speed of $c$ also seems to be a problem. i don't think it's accurate to say it has nothing to do with it. there are different ways to look at it (that lead to the same conclusion).

it's pretty hard to see something approaching infinite inertial mass to accelerate further.
An object accelerates from its rest frame, not from some other frame.

There are an infinite number of speeds that an object is traveling at when measured from an infinite number of frames of reference. But all these are completely irrelevant.

Right now, you and I are traveling at a speed very close to c in some frame of reference. Would we care for a minute that that is the case, and would it care if we accelerate?

If you accelerate for 10 million years you would still be at rest in your rest frame. And if you then accelerate even more, the prior 10 million years of accelerations would have no influence on it at all, it is as if you accelerate for the first time.

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pervect
Staff Emeritus
I don't understand. What is "E_relative" and how do you do to measure it? If a concrete example is needed then use an electron as an example.

Best regards

Pete

Just measure the kinetic energy of the electron, and plot momentum vs kinetic energy. Done at high enough energies, fitting to this curve will tell you about the rest mass of the electron (though perhaps not with high accuracy).

Also, experimentally, one can compare this number to the number one get from the above technique to the energy emitted by gamma rays from electron-positron annhiliation.

JesseM
An object accelerates from its rest frame, not from some other frame.
What is the physical meaning of this statement? I think you're introducing your own vocabulary again, there isn't any standard meaning of the phrase "accelerates from" that would make this true. The relativistic mass does tell you the object's resistance to acceleration in its direction of motion as seen in a frame where it's moving, in terms of the force or energy that must be applied in that frame (not the rest frame) to increase the object's velocity by a small increment. So, saying "you can't accelerate an object to light speed in your frame because its relativistic mass would go to infinity" is equivalent to the statement "you can't accelerate an object to light speed in your frame because the energy you'd need to apply to accelerate it (as measured in your frame) would go to infinity", which is a reasonable way of explaining why it's impossible to accelerate to light speed.

What is the physical meaning of this statement? I think you're introducing your own vocabulary again, there isn't any standard meaning of the phrase "accelerates from" that would make this true.
Acceleration is absolute not relative.
The accelerating object semi-rotates in space time, and while the change in "angles" is relative, the "movement" is absolute.

The relativistic mass does tell you the object's resistance to acceleration in its direction of motion as seen in a frame where it's moving, in terms of the force or energy that must be applied in that frame (not the rest frame) to increase the object's velocity by a small increment.
Sure, but that is simply a Lorentz transformation. Only Lorentz invariant quantities are the same in all frames!

So, saying "you can't accelerate an object to light speed in your frame because its relativistic mass would go to infinity" is equivalent to the statement "you can't accelerate an object to light speed in your frame because the energy you'd need to apply to accelerate it (as measured in your frame) would go to infinity", which is a reasonable way of explaining why it's impossible to accelerate to light speed.
It may be equivalent but it is a wrong argument.
And even if you apply unlimited energy you still won't reach c!

This "infinite energy argument" is very common but it is wrong.

The only way you can approach c in the limit is by reducing your mass until it is zero.

The reason you won't reach c in the limit is purely kinematic and directly related to the hyperbolic nature of a hypersurface in space-time.

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JesseM
Acceleration is absolute not relative.
Proper acceleration is absolute, coordinate acceleration is not. And again, there is no standard meaning to the phrase "accelerates from" that would make your statement "An object accelerates from its rest frame, not from some other frame" correct.
MeJennifer said:
It may be equivalent but it is a wrong argument.
What specifically is wrong? Do you disagree that relativistic mass tells you the amount of energy you must apply in your frame to increase an object's coordinate velocity by a small increment? Do you disagree that the relativistic mass going to infinity means that the amount of energy you'd need to apply in your frame to increase an object's coordinate velocity to c would be infinite?
MeJennifer said:
The reason you won't reach c in the limit is a purely kinematic
Is your kinematic argument that accelerating to the speed of light is impossible because the proper acceleration would need to be infinite? Without some additional assumptions about the force or energy needed to change an object's proper acceleration, the argument seems incomplete--the infinite proper acceleration might just be a sort of coordinate artifact, you haven't shown that infinite proper acceleration is actually a physical impossibility.

You do not seem to understand what I say JesseM.

Take the velocity addition formula and find the limit if both u and v approach c. You will see that there is no such limit.

So if you accelerate an infinite number of years with an infinite number of g you still don't reach c, not for the restframe and not from any other frame of reference.

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JesseM
Take the velocity addition formula and find the limit if both u and v approach c. You will see that there is no such limit.
Again, without some additional arguments about energy, you can't rule out the possibility that this is a coordinate artifact. We already know that things moving at c cannot have their own inertial frame, so any attempt to figure out how things look "from their perspective" gives nonsensical results, but this doesn't mean we can't see things moving at c in our own frame.
MeJennifer said:
So if you accelerate an infinite number of years with an infinite number of g you still don't reach c, not for the restframe and not from any other frame of reference.
Do you agree that if I see some object experiencing a certain coordinate acceleration in my frame, I can figure out the corresponding proper acceleration it's experiencing? Do you agree that we could write down some formula for coordinate acceleration as a function of coordinate time in my frame which would cause the coordinate velocity to reach c at some finite time $$t_c$$ (making no assumptions about whether this is physically realistic or unrealistic a priori), and that if we calculate the proper acceleration as a function of coordinate time, it approaches infinity in the limit as the coordinate time approaches $$t_c$$? It might be that the proper time experienced by the ship also goes to infinity in this limit, I'm not sure, but in any case I'm just interested in the question of whether it's possible or impossible for me to see something accelerate to c in a finite time in my frame, and without bringing in some arguments about the energy I don't see how you can prove this is impossible.

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I give up

I give up!

Keep thinking that if you apply an infinite amount of energy or if your acceleration is infinite then you travel at c from some other frame of reference.

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JesseM
Again, one can define some function for coordinate velocity as a function of coordinate time in my frame, like v(t) = a*t, which will cause the coordinate velocity to reach c at some finite coordinate time $$t_c$$. And it would actually be physically possible to cause a real object to move like that up to any finite time $$t_c - \epsilon$$ before $$t_c$$. I could do this by firing a laser of ever-increasing energy at the back of the object to push it along, for example. The point is that in the limit as you get arbitrarily close to $$t_c$$ (as $$\epsilon$$ is made arbitrarily small), the energy required to accelerate the object this way (as measured in my frame) becomes arbitrarily large--surely you don't disagree with this? This is all I'm saying, and please note that the argument is solely concerned with quantities measured in my frame (the energy in my frame, the coordinate velocity, and the coordinate time), it's irrelevant to the argument what proper acceleration or proper time is experienced by the object itself. If the energy required in my frame to push an object up to c were finite, then I don't see that accelerating an object to c could be physically ruled out, even if you got infinite (or undefined) answers for the proper acceleration and/or proper time when you tried to figure out how this motion would look from the object's point of view; again, you'd be free to dismiss these answers as a coordinate artifact, much as we dismiss the results of plugging v=c into the Lorentz transformation, and don't take the infinite answers to imply that it's impossible for anything to move at c. Without bringing in the issue of energy, you haven't explained why the undefined/infinite answers in the case of accelerating to c couldn't be dismissed as a coordinate artifact in the same way.