Mass Energy: Understanding Total, Rest and Kinetic Energies

[QuantumKing]

I have a hard time understanding the concepts of total energy, rest energy and kinetic energy.

I know that when a mass is at rest, it is moving at c through spacetime. Motion through time can be diverted into motion through space, resulting in a "slower" time, relative to another inertial system...Anyways, my question is, why is the total energy of a system equal to its rest energy + its kinetic energy? Wouldnt its energy at rest be converted into kinetic energy, which is the energy of motion?..I know that a body in motion gains mass, and with e=mc^2 that would mean the total energy would increase, but I am thinking that c would decrease to conserve the energy of the body previoulsy at rest. Since motion through time is diminished when motion through space is taking place, how come this isn't the case??

Some help with this would be greatly appreciated:P thanks

Eric

 

[pmb_phy]

I have a hard time understanding the concepts of total energy, rest energy and kinetic energy.

I know that when a mass is at rest, it is moving at c through spacetime.

That is a comment that people love to say and for which it has no meaning. The term "speed" as in "speed of light c" means that an object has moved a certain about of "space" during a certain amount of "time". Since this cannot be applied to the term "speed" as applied to "moving through spacetime" it is a nonsense phrase to be avoided like the plauge.

Anyways, my question is, why is the total energy of a system equal to its rest energy + its kinetic energy?

The answer is available through calculation. But the total energy is the sum of the rest energy, kinetic energy and potential energy. For a derivation please see

http://www.geocities.com/physics_world/sr/relativistic_energy.htm

Wouldnt its energy at rest be converted into kinetic energy, which is the energy of motion?

No. Why would you think so?? Kinetic energy is energy of motion, potential energy is energy of position. Rest energy is energy of rest mass.

Best wishes

Pete

 

[actionintegral]

Rest energy is not "converted into" kinetic energy. It is retained by the object. The kinetic energy is then added to that.

 

[lightarrow]

Wouldnt its energy at rest be converted into kinetic energy, which is the energy of motion?

This can happen, sometimes and for a little percentage of the rest mass. Examples:
-chemical reactions releasing energy (Very little percentage of rest mass transformed into energy, a part of which can be transformed into kinetic energy).
-nuclear reactions (about 0.1%)
matter-antimatter annichilation (all the rest mass becomes energy, a part of which...).

 

[masudr]

That is a comment that people love to say and for which it has no meaning. The term "speed" as in "speed of light c" means that an object has moved a certain about of "space" during a certain amount of "time". Since this cannot be applied to the term "speed" as applied to "moving through spacetime" it is a nonsense phrase to be avoided like the plauge.

Yes indeed. What they really mean is that the 4-velocity vector has only a non-zero time component when the object is at rest. If it has a velocity v (or we are in a ref. frame moving at -v), then the 4-velocity vector starts pointing in these other directions: the component in the time direction gives us the Lorentz contraction of time.

Of course, the real reason for this is that 4-velocity is a spacetime vector, as is 4-position, and not the notions of speed through spacetime, as you have explained. I blame Brian Greene for this section in his book The Elegant Universe which expounds this description.

 

[actionintegral]

Examples:
-chemical reactions releasing energy (Very little percentage of rest mass transformed into energy, a part of which can be transformed into kinetic energy).

Can you tell me how this happens? I need to know "which" rest mass gets converted into energy

 

[lightarrow]

Can you tell me how this happens? I need to know "which" rest mass gets converted into energy

I think you are talking about potential energy, isnt'it?
If this is the case, I agree with the fact that what is rest mass for molecules is, partially, potential energy of atoms, nuclei and electrons.
However, the rest mass of a nucleus is, partially, potential energy of nucleons...and so on. Maybe everything is potential energy.

In chemistry, we start from the standard state of reagents to arrive to the product's standard states. Example: molecular hydrogen + molecular oxygen--> molecular water. The difference in mass from products and reagents has gone away as energy.

So, it's (part of) the rest mass of molecules which gets converted into energy.

 

[actionintegral]

I think I see. Let me ask this question then: Suppose I have two identical objects. One object is at rest, floating in space, and the other is at rest, but at the end of a spring that I stretched out (so it has potential energy).

Please compare the relativistic masses of the two objects.

 

[pervect]

The stretched out object will have a higher "relativistic mass". It will also have a higher "invariant mass".

I generally prefer to deal with invariant mass, because it is the only sort of mass that is a property of the system and independent of the observer, as long as the system is isolated. Non-isolated systems have several tricky aspects to them. Unfortunately, your example is a non-isolated system, unless you let the mass on the end oscillate naturally. If you hold it in a "stretched out" position, you do not have an isolated system.

Invariant mass is, in geometric units, E^2 - p^2, where E is the energy of the system, and p is its momentum.

When you stretch out the spring, you increase E, but you do not change p, which remains at zero. Ergo, you increase the invariant mass.

Relativistic mass is almost universally understood as being equivalent to energy, so when you increase the energy E, you also increase the relativistic mass.

Note that, except for gravitational radiation (negligible), if your mass-spring system is isolated, the mass stays constant. The spring converts potential energy into kinetic energy, and vica-versa, but the total energy of the system stays constant. This means the relativistic mass stays constant, and because the momentum stays constant (and zero), the invariant mass is also constant.

You might try the wiki articles:

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

 

[actionintegral]

Hi Pervect,

I read your answer carefully, but I am confused. You refer to an attribute of the object called "invariant mass", but then you say that it increases.

Furthermore, the object is still at rest, regardless of whether it is attached to a spring or not.

I am trying to see if potential energy can be considered as relativistic mass. From where I stand, it doesn't seem like it.

 

[QuantumKing]

so, back to my first question, apparently brian greene is a dueshbag? lol, so, what he said about motion through time being diverted into motion through space is just a load of crap? Although it made sense to me, explaining the time dilation phenomena..so what did einstein base his equation of rest energy on? why does he use the constant c?

 

[actionintegral]

I am familiar with that idea. The concept was not "motion through time" per se but rather a trade-off between velocity and proper-time.

The concept of rest energy is based on the relativistic definition of work.

 

[actionintegral]

QUOTE:
Originally Posted by lightarrow
Examples:
-chemical reactions releasing energy (Very little percentage of rest mass transformed into energy, a part of which can be transformed into kinetic energy).
QUOTE


Please check the following for correctness:

An object of mass m0 is at rest and attached to a spring.

We move the object and stretch the spring.

Then we stop the object, and the spring remains stretched.

At this time, the mass of the object is still m0.

Therefore, it is incorrect to refer to the potential energy of the loaded spring as mass. Perhaps you could call it "potential mass".

 

[jtbell]

You [pervect] refer to an attribute of the object called "invariant mass", but then you say that it increases.

"Invariant" in this context means "invariant between inertial reference frames." That is, the invariant mass of an object (in a given state) is the same when measured by different observers moving at constant velocity with respect to each other. It does not mean "invariant between its stretched and unstretched conditions."

 

[actionintegral]

It does not mean "invariant between its stretched and unstretched conditions."

Understood. What started the thread was the suggestion that chemical reactions convert matter to energy. I don't get that.

 

[jtbell]

Let's first look at nuclear reactions rather than chemical reactions. If a nucleus can spontaneously decay into another one plus some other particles, releasing energy in the process, then the sum of the invariant masses of the final nucleus and the other particles is less than the invariant mass of the original nucleus, and the difference corresponds the the energy released, via $E = mc^2$. We know this is true, because we can measure nuclear and particle masses with enough precision to verify it.

We assume that the same sort of thing is true for chemical reactions. That is, we expect that in an exothermic reaction (one that generates heat), the sum of the invariant masses of the product molecules is less than the sum of the invariant masses of the reactant molecules, with the difference corresponding to the heat relased. I don't think this has been experimentally verified (although I'd be delighted to find out that it has), simply because the mass difference is exceedingly tiny.

Does this mean that matter is converted to energy? It depends on whether you consider "matter" to be equivalent to "invariant mass". The invariant mass of a composite system such as a molecule, atom or nucleus, is the sum of the invariant masses of its components, plus the mass-equivalent of the potential energy associated with binding the system together, plus the kinetic energy of the "internal motion" of the components (more precisely, the kinetic energy of the components in a reference frame where the system as a whole is at rest, i.e. the total momentum is zero).

Do you consider the part of a molecule's invariant mass that is due to potential energy or internal kinetic energy of its atoms, to be "matter"? If yes, then you can consider an exothermic chemical reaction as converting matter into energy. Otherwise... well, you can fill that in for yourself.

 

[pervect]

Hi Pervect,

I read your answer carefully, but I am confused. You refer to an attribute of the object called "invariant mass", but then you say that it increases.

Furthermore, the object is still at rest, regardless of whether it is attached to a spring or not.

I am trying to see if potential energy can be considered as relativistic mass. From where I stand, it doesn't seem like it.

Please read the wiki links I quoted for some more of the background information if you are not familar with invariant mass.

I'll repeat the most important link
http://en.wikipedia.org/wiki/Rest_mass

You might also try
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

I thought my answer was reasonably clear - "relativistic mass" is another name for energy, and when you stretch a spring, you increase the energy of the spring, therefore you increase its relativistic mass.

Relativistic mass depends on the observer, as I commented before. Invariant mass does not. If you read the wiki articles I mentioned, they go into a lot more detail, including the defintions, which can be summarized as

m_r = E/c^2
m_i = sqrt(E^2 - (pc)^2)/c^2

where E is the energy of a system and p is its momentum. You can see that if p=0, the two defintiosn are equivelant, i.e. relativistic mass equals invariant mass in the "rest frame" of a system, the frame in which the momentum of a system is zero.

I would suggest reading the links I quoted for more of the background information on the defintions of "relativistic mass" and "invariant mass" if these terms aren't familiar.

 

[actionintegral]

"relativistic mass" is another name for energy

I would agree with you if you replace the word "energy" with "kinetic energy" in the above phrase.

 

[Garth]

I would agree with you if you replace the word "energy" with "kinetic energy" in the above phrase.

Actually it should be "total energy" - that due to rest mass or the energy-momentum of the object, plus "kinetic energy".

Garth

 

[actionintegral]

Hi Garth,

The reason I have focused on this point is I am trying to understand why
some people refer to potential energy as mass. What is confusing me is that potential energy is a mathematical fiction (what would happen IF I let go...) whereas kinetic energy is quite measurable.

 

[lightarrow]

Hi Garth,

The reason I have focused on this point is I am trying to understand why
some people refer to potential energy as mass. What is confusing me is that potential energy is a mathematical fiction (what would happen IF I let go...) whereas kinetic energy is quite measurable.

Try to take with your hands the two poles of a 400,000 volts power line, and you will understand if potential energy is measurable!

Of course you could argue that those are just the "effects" of potential energy; I think it's only a matter of words. What is energy? "Just" the "possibility" to make work. So, even kinetic energy is "just a possibility of..." Does always kinetic energy have an interaction with bodies? No. If two balls collides elastically, for example, kinetic energy don't do just anything, as potential energy that is not released.

 

[jtbell]

If potential energy is a mathematical fiction, why is the (invariant) mass of an atomic nucleus less than the sum of the individual masses of its constituent protons and neutrons, if not because of the negative potential energy?

 

[actionintegral]

If potential energy is a mathematical fiction, why is the (invariant) mass of an atomic nucleus less than the sum of the individual masses of its constituent protons and neutrons, if not because of the negative potential energy?

Thank you for answering my question. That's very interesting.

 

[Jorrie]

I would agree with you if you replace the word "energy" with "kinetic energy" in the above phrase.

Since Pervect has not answered (yet), I'll just point out this: kinetic energy is not the same thing as 'relativistic mass'. If you take the rest mass of an object and add kinetic energy (but no other energy) to it, you have it's total energy, or 'relativistic mass'. So I think Pervect is absolutely correct.

 

[pmb_phy]

Since Pervect has not answered (yet), I'll just point out this: kinetic energy is not the same thing as 'relativistic mass'. If you take the rest mass of an object and add kinetic energy (but no other energy) to it, you have it's total energy, or 'relativistic mass'. So I think Pervect is absolutely correct.

relativistic mass is not another name for energy. They aren't even proportional in all cases. E = mc^2 holds only in the case of an isolated system, not for non-isolated systems. An example is given at

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Pete

 

[pmb_phy]

Hi Garth,

The reason I have focused on this point is I am trying to understand why
some people refer to potential energy as mass. What is confusing me is that potential energy is a mathematical fiction (what would happen IF I let go...) whereas kinetic energy is quite measurable.

I disagree. Potential energy is just as real as kinetic energy. In the last analysis each form of energy is measured by measuring certain dynamical quantities such as position, speed and proper mass. The total energy is then found by adding these numbers. Energy is a pretty abstract thing which so far has been undefinable. I.e. nobody knows what energy is, just how to measure it in all its forms.

Pete

 

[Jorrie]

relativistic mass is not another name for energy. They aren't even proportional in all cases. E = mc^2 holds only in the case of an isolated system, not for non-isolated systems. An example is given at

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Pete

Ok Pete, I'll rest my case on total energy and 'relativistic mass' for now, but I don't accept it as settled (yet). I only wanted to point out that 'kinetic energy' is not the same thing as 'relativistic mass', which I do not think you will have an issue with.

 

[pervect]

relativistic mass is not another name for energy. They aren't even proportional in all cases. E = mc^2 holds only in the case of an isolated system, not for non-isolated systems. An example is given at

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

Pete

Pete says this a lot, but his usage does not appear to be standard in this particular area.

For instance, consider Griffiths, "Introduction to Electrodynamics"

Einstein called

m_rel = m / sqrt(1-u^2/c^2)

the relativistic mass (so that p⁰ = m_relc and p = m_relv; m itself was then called the rest
mass), but modern usage has abandoned this terminology in favor of relativistic energy

Also, older books, such as Tollman's "Relativity, thermodynamics, and cosmology" also define relativistic mass in terms of energy. Being older books, they don't call it "relativistic mass", but simply "mass", while modern usage is to call this "relativistic mass".

And in addition we have already seen in 23 that we must ascribe mass to
the potential energy generated during the course of an elastic collision
(see 23.7). Hence in what follows we shall postulate in general that a
quantity of energy E always has immediately associated with it a mass
of the amount E = m/c^2.

In addition, as a further consequence of the association of mass with
any given quantity of energy, as given by equation (27.3), it would
also appear natural to assume the reciprocal relation of an association
of energy with any given quantity of mass.

Such defintions are also used in the sci.physics.faq.

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

which defines m_r as E/c^2.

I have never personally seen a reference that uses the same usage that Pete does, and I've just quoted three that use the term the way I use it.

It is possible that such references (using Pete's idea) exist, but this makes the concept of relativistic mass ambiguous at best, because it is clear from the two references that I've quoted that relativistic mass is often considered to be a synonym for energy. In fact, from what I've seen, defining relativistic mass as energy/c^2 is the more common and standard usage, and Pete seems to be using the term in a very personal and non-stadard way.

I gather that Einstein first introduced the concept in 1907, while I can find what is probably the German original I don't read German. (The original is in the wiki version of the collected works of Einstein, the 1907 paper about energy looks promising. The 1907 date came from the sci.physics.faq.)

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

As the issue applies only to systems with a non-zero volume, I have some doubts that the interpretation of what Einstein meant will be all that clearcut from what he wrote - I strongly suspect that his paper only talked about point particles, and that Einstein never even considered the issue.

 

[pervect]

Hi Garth,

The reason I have focused on this point is I am trying to understand why
some people refer to potential energy as mass. What is confusing me is that potential energy is a mathematical fiction (what would happen IF I let go...) whereas kinetic energy is quite measurable.

Potential energy isn't quite a mathematical fiction. You might want to consider a simpler problem to see why.

Suppose you have two charged spheres, one with a charge of +q and another with a charge of -q. They attract each other, like your spring. When the two charges are closer together, they have less energy, when they are further apart, they have more energy.

So how is energy conserved? The answer is that energy is conserved because the electric field, itself, has energy. When you change the spacing between spheres, you change the electric field.

The density of energy in the electric field is .5 epsilon E^2

See http://hyperphysics.phy-astr.gsu.edu/hbase/electric/engfie.html

So when you add the energy stored in the fields, you can see that potential energy is not "fictional", its just stored in a field.

Your spring example is more complicated, but basically the energy stored in a spring gets stored in the electromagntic fields which hold the atoms in a solid together.

 

[pmb_phy]

Ok Pete, I'll rest my case on total energy and 'relativistic mass' for now, but I don't accept it as settled (yet). I only wanted to point out that 'kinetic energy' is not the same thing as 'relativistic mass', which I do not think you will have an issue with.

Hi Jorrie

I don't know where you got the impression that I believe that relativistic mass is proportional to kinetic energy. And I never say that 'relativistic mass' is energy. If you have Rindler's 1982 intro to SR text he states that explicitly. Rindler is where I picked up on this not to well known fact.

Pete

 

[pmb_phy]

And mass isn't condensed energy.

Ah! Yes! Someone after my own heart. I've been saying that for a very long time. A simple countrer example is a box with a gas in it. Nothing is condensed here. The greater the kinetic energy of the gas particles the greater the mass of the box-gas system. But there is nothing condensed here to think about. And when a positron collides with an electron then its clear that it will emit radiation. However since the radiation has (relativistic mass) then the total mass of the system remains unchanged. The only converrsion going on here is the form the mass takes on.

Pete

 

[lightarrow]

As long as we talk about relativistic mass and total energy, mass and energy are equal, excepting for c^2 factor. So, in this case, it's meaningless to talk about "mass is condensed energy", since they are the same thing.

If we talk about rest mass, then it's different.

IMHO, If the photon is spatially localized with approximated dimensions greater than those of the electron and positron, we can pheraps say the energy is more "condensed" in the particles than in the radiation: in the particles energy is stored in the form of rest mass (mostly); in the photon is present in the form of electromagnetic radiation (no rest mass).

So, as pmb_phy says, it's only the form of the energy that changes.

 

[Jorrie]

Hi Jorrie

I don't know where you got the impression that I believe that relativistic mass is proportional to kinetic energy. And I never say that 'relativistic mass' is energy. If you have Rindler's 1982 intro to SR text he states that explicitly. Rindler is where I picked up on this not to well known fact.

Pete

Hi Pete,

I was originally referring to Actionintegral's reply to Pervect:

Originally Posted by pervect: "relativistic mass" is another name for energy

and Actionintegral's reply:

"I would agree with you if you replace the word "energy" with "kinetic energy" in the above phrase."

So my reply to yours was slightly confusing if the context was not clear.

But hey Pete, I think you missed that original statement by Pervect!

Regards, Jorrie

 

[actionintegral]

This is why I steer clear of questions with lots of responses!

 

[Farsight]

Ah! Yes! Someone after my own heart...

Thanks Pete. I do think loose phrases like "energy consumption" or "pure energy" cause problems for the man in the street.