# Why Mass Doesn't Always Add When Two Systems Combine

• pivoxa15
In summary: Space isotropy means that the directions in space are the same for all observers. Time homogeneity means that the order of events is the same for all observers.
pivoxa15
Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?

Energies and momentum add because of conservation laws but mass is conserved as well isn't it?

Or is it the case that we are dealing with invariant mass hence is special? This sounds a bit mystical.

Noether theorem.

E and p are components of a 4-vector p^\mu=(E;p).
m=\sqrt{E^2-p^2}. Why would you think it should add?

pivoxa15 said:
Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?

Energies and momentum add because of conservation laws but mass is conserved as well isn't it?

Or is it the case that we are dealing with invariant mass hence is special? This sounds a bit mystical.
Please give an example where mass doesn't add, while energy does. If an electron and a positron annihilate into radiation, no rest-mass is left over but energy is radated away. Now, this radiated energy also represents mass (by m=E/c^2) and so the total mass (but not rest-mass) is always conserved.

notknowing said:
Please give an example where mass doesn't add, while energy does. If an electron and a positron annihilate into radiation, no rest-mass is left over but energy is radated away. Now, this radiated energy also represents mass (by m=E/c^2) and so the total mass (but not rest-mass) is always conserved.

Note that pivoxa15 used the term "invariant mass" in the original post. Also, although Pete disagrees, over the last several decades, the unmodified term "mass" has almost universally come to mean rest mass.

Invariant mass is the "magnitude" of the energy-momentum four-vector. When you add four-vectors in general, the magnitudes don't usually add, just like the magnitudes of ordinary three-vectors don't add when they're in different directions.

jtbell said:
Invariant mass is the "magnitude" of the energy-momentum four-vector. When you add four-vectors in general, the magnitudes don't usually add, just like the magnitudes of ordinary three-vectors don't add when they're in different directions.

This is a nice logical argument for why invariant mass dosen't add.

I also realized that the conservation law applies for relativistic mass, energy, momentum only. The invariant mass is not relativistic hence not conserved.

pivoxa15 said:
Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?
Once again we need to be precise as to what is meant by mass. If you mean inertial mass (aka relativistic mass) then yes. Inertial mass adds. In fact it is inertial mass that is the time component of 4-momentum. To see that inertial mass is conserved I've worked out a proof. Please see

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

Energies and momentum add because of conservation laws but mass is conserved as well isn't it?
Inertial mass, yes. Proper mass, no.

Or is it the case that we are dealing with invariant mass hence is special?
That is what it sounds like to me too.

robphy said:
Noether theorem.
Um ... In order for the OP to use that law they must know what the term "mass" means in order to apply the law. The only result that will yield a conserved quantity from Noether's theorem is "relativistic mass." Is that what you meant??

Meir Achuz said:
E and p are components of a 4-vector p^\mu=(E;p).
m=\sqrt{E^2-p^2}. Why would you think it should add?
The assertion that 4-vectors add is a correct one and is fundamental to tensor analysis. In curved spacetime the 4-vectors to be added must be located at the same event. In flat spacetime they need not be located at the same event. For details please see

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

Best wishes

Pete

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pivoxa15 said:
The invariant mass is not relativistic

Eh?

Can you expand on this? I have the feeling that some other word than "relativistic" might be better here. Invariant quantities are an important part of SR.

Noether's Theorem is beuatiful, it gives a hint to the underlying symmertry behind Space and Time before SR!

energy and momentum r conserved because they r energetic forms, whereas mass is a form of condensed form of it.mass is not pure energy.it is not quantized

jtbell said:
Eh?

Can you expand on this? I have the feeling that some other word than "relativistic" might be better here. Invariant quantities are an important part of SR.

I forgot one word, mass. And probably should include a special case as well.

Here is the complete sentence. 'The invariant mass is not relativistic mass (for nonzero kinetic energy) hence not conserved.'

Unfortunately, Noether's theorem looks a bit too technical for me.

pivoxa15 said:
Unfortunately, Noether's theorem looks a bit too technical for me.
I don't know which are your mathematical bases, otherwise I think I could explain you.
In simple words it comes out from the Stationary Action Principle that:

1. Space homogeneity --> linear momentum is conserved
2. Space isotropy --> angular momentum is conserved
3. Time homogeneity --> energy is conserved

Space homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been put in another place of space.

Space isotropy means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been rotated in space.

Time homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments in another moment of time.

Unfortunately all this is not true in general in curved space-time (if space-time is curved, there cannot be homogeneity or isotropy!).
So, in general, in a curved space-time, energy, linear momentum and angular momentum are not conserved. They are not even simple to define.

pivoxa15 said:
Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?

Energies and momentum add because of conservation laws but mass is conserved as well isn't it?

Or is it the case that we are dealing with invariant mass hence is special? This sounds a bit mystical.
If mass would always add, than we would be living in a boring universe. In that case, radiation/ light could never be emitted in a nuclear or chemical reaction and the universe would be completely dark...

notknowing said:
If mass would always add, than we would be living in a boring universe. In that case, radiation/ light could never be emitted in a nuclear or chemical reaction and the universe would be completely dark...

Could you give reasons for this claim?

lightarrow said:
I don't know which are your mathematical bases, otherwise I think I could explain you.
In simple words it comes out from the Stationary Action Principle that:

1. Space homogeneity --> linear momentum is conserved
2. Space isotropy --> angular momentum is conserved
3. Time homogeneity --> energy is conserved

Space homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been put in another place of space.

Space isotropy means that the results of experiments inside a laboratory don't change if you perform those experiments with the laboratory which has been rotated in space.

Time homogeneity means that the results of experiments inside a laboratory don't change if you perform those experiments in another moment of time.

Unfortunately all this is not true in general in curved space-time (if space-time is curved, there cannot be homogeneity or isotropy!).
So, in general, in a curved space-time, energy, linear momentum and angular momentum are not conserved. They are not even simple to define.

I know as much maths as a 2nd year Uni student.

Given that we live in a curved space-time, Noether's principle doesn't hold in reality? Also what is conserved in reality, that is in a curved space-time?
Reality as in reality as we know it today offcourse.

Pivoxa15:
So, probably you are able to read this document about Noether's Theorem:

http://www.mathpages.com/home/kmath564/kmath564.htm

where you only need to know (if you don't already) that the action S is the integral between two instants of time t_1 and t_2 of the Lagrangian L = T-V where T is kinetic energy and V the potential energy, and the variation deltaS is made with fixed values of q_i(t) at the two instants t_1 and t_2.

About your first questions: far from massive objects (e.g. stars, black holes), the curvature is negligible, so our space-time can be considered flat;

Think that not even the velocity vector v has a unique global meaning (it depends on how it is transported from one point to another).

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pivoxa15 said:
Could you give reasons for this claim?
The only way to make the mass of the combined system equal to the sum of the separate masses is to ensure that nothing (no energy) escapes during the process of combining. So, it means that no energy/light can be radiated away. I'm not sure how to explain this better.

notknowing said:
The only way to make the mass of the combined system equal to the sum of the separate masses is to ensure that nothing (no energy) escapes during the process of combining. So, it means that no energy/light can be radiated away. I'm not sure how to explain this better.
You only need to be clear about what the system is that you're determining the conservation of mass. If the system is closed then the total inertial mass is a constant in time. There was confusion on this issue in a newsgroup once. Someone was making bogus claims and saying that inertial mass (aka relativistic mass) was not conserved. I posted the correction here

http://www.geocities.com/pmb_phy/bogus_claims.htm

See the part entitled Bogus Claim No. 3 - The inertial mass (aka relativistic mass) of a closed system is not conserved.

I address a case similar that you mention. I.e. a pion decaying into two photons.

Best wishes

Pete

pivoxa15 said:
Can someone provide a simple and convincing reason why mass dosen''t always add when two system combine but momentum and energy always does?

Energies and momentum add because of conservation laws but mass is conserved as well isn't it?

Or is it the case that we are dealing with invariant mass hence is special? This sounds a bit mystical.
should we mention if the considered systems interact or not?
sine ira et studio

George Jones said:
Also, although Pete disagrees, over the last several decades, the unmodified term "mass" has almost universally come to mean rest mass.
What is is that you believe that I disagree with? If its usage that you speak of then here is my observation

(1) Particle physicists use "mass" to mean "proper mass." They do so because this is the property that they study and to keep qualifying it with "proper" would be just plain monotonous. Even I wouldn't do it.

(2) If you're speaking about what appears in current SR/GR textbooks then its varied. It seems close to half and half.

(3) If you're referring to texts on things like cosmology then it appears that when "mass" is used unqualified then it means "inertial mass." However in most textbooks that I know of the authors are aware of this debate and they qualify the term when they first use it.

(4) As for the literature - Depends on the topic of the article. I would hazard to guess that 10% use "mass" to mean "inertial mass."

Since I haven't gone through the last 20 years of all the physics literature I am unable to give you exact statistics. However many online files, such as information files for workers at CERN do use mass to mean inerital mass.

Beyond what I just said here would be someone putting words in my mouth. As to what I believe should be used as far as what "mass" means, ... well who really cares?

Best wishes

Pete

## 1. Why doesn't the mass always add up when two systems combine?

When two systems combine, the total mass may not always be equal to the sum of the individual masses. This is because mass is not a conserved quantity in all systems. In some cases, mass may be converted into other forms of energy, such as kinetic or potential energy, which can result in a decrease in the total mass. Additionally, the concept of mass can be different in different reference frames, leading to discrepancies in the total mass when two systems are combined.

## 2. What factors can affect the total mass when two systems combine?

There are several factors that can affect the total mass when two systems combine. These include the conversion of mass into other forms of energy, the different definitions of mass in different reference frames, and the presence of external forces, such as gravity or electromagnetic forces, which can alter the mass of the combined system.

## 3. Is the law of conservation of mass violated when two systems combine?

No, the law of conservation of mass is not violated when two systems combine. This law states that the total mass in a closed system remains constant over time. In cases where the total mass of two combined systems is less than the sum of the individual masses, this can be explained by the conversion of mass into other forms of energy, which does not violate the law of conservation of mass.

## 4. Can the mass of a combined system ever be greater than the sum of the individual masses?

Yes, in some cases, the mass of a combined system can be greater than the sum of the individual masses. This can occur when the combined system gains mass through the addition of external particles, such as in a chemical reaction, or when the combined system is affected by external forces that increase its mass, such as in a nuclear reaction.

## 5. How can we accurately measure the mass of a combined system?

Accurately measuring the mass of a combined system can be challenging due to the various factors that can affect the total mass. However, scientists use a variety of techniques and instruments, such as mass spectrometry and gravimetry, to accurately measure the mass of a combined system. These methods take into account any changes in mass due to conversion into other forms of energy or the influence of external forces, providing a more precise measurement of the total mass of the combined system.

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