# Prove e=mc^2

shivaniits
if possible could you tag some good video lectures may be feymann or any other good source..thanks...!!

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Whovian

Prove? This is like asking someone to prove Quantum Mechanics. Mathematically, there are a ridiculous number of consistent theories, so all we can do is fail to disprove it.

shivaniits

are there any good feymann lectures on special theory of relativity or quantum mechanics...???
thanks!!

shivaniits

@mathman thanks i will try them!!

haael

In special relativity, time is just another dimension, just like spatial distance. The numerical value of the speed of light is a proportionality factor between time and length units. This identification causes a series of other identifications.

Speed becomes an angle between the time directions between two moving reference frames. This also nicely explains Lorentz transformations and relativistic speed composition.

Mass, momentum and energy are also identified. The proportionality factors between the standard units of these parameters are: momentum is mass times c and energy is mass times c squared. That's all.

Electric and magnetic potentials also undergo an unification into electromagnetic potential.

This is the basic idea of the special relativity. We live in a 4D space. Our common 3D physical properties turn out to be spatial or time-like components of 4-dimensional properties. Time and space are the same, but our standard unit system (say SI system) has different units for them. When we want to convert from time to space, we need to multiply the numerical value of the property by some proportionality factor, which is always some power of c, depending on the dimesionality of our property.

shivaniits

In special relativity, time is just another dimension, just like spatial distance. The numerical value of the speed of light is a proportionality factor between time and length units. This identification causes a series of other identifications.

Speed becomes an angle between the time directions between two moving reference frames. This also nicely explains Lorentz transformations and relativistic speed composition.

Mass, momentum and energy are also identified. The proportionality factors between the standard units of these parameters are: momentum is mass times c and energy is mass times c squared. That's all.

Electric and magnetic potentials also undergo an unification into electromagnetic potential.

This is the basic idea of the special relativity. We live in a 4D space. Our common 3D physical properties turn out to be spatial or time-like components of 4-dimensional properties. Time and space are the same, but our standard unit system (say SI system) has different units for them. When we want to convert from time to space, we need to multiply the numerical value of the property by some proportionality factor, which is always some power of c, depending on the dimesionality of our property.
so can we say this conversion of E=mc^2 doesn't happen at all
because it what we perceive from different reference frames due to account of special relativity
and we can't say that
actually a some lump let say of m mass is vanishing(or more precisely should i say converting to energy)... doesn't really happen whatever we measure is due to special relativity effect!!

so can we say this conversion of E=mc^2 doesn't happen at all
because it what we perceive from different reference frames due to account of special relativity
and we can't say that
actually a some lump let say of m mass is vanishing(or more precisely should i say converting to energy)... doesn't really happen whatever we measure is due to special relativity effect!!

Utter nonsense! Pair production converts photons to electron-positron pairs. Nuclear fission or nuclear fusion convert mass to energy.

Staff Emeritus
Gold Member

Prove? This is like asking someone to prove Quantum Mechanics. Mathematically, there are a ridiculous number of consistent theories, so all we can do is fail to disprove it.

The OP is asking whether we can prove it from more fundamental principles that have been established experimentally. The answer is yes, and that's exactly what Einstein did in his 1905 paper "Does the inertia of a body depend upon its energy content?," http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

haael
Pair production converts photons to electron-positron pairs. Nuclear fission or nuclear fusion convert mass to energy.
You speak about converting rest mass to kinetic energy. This is indeed a physical process.

OP was refering to total mass and total energy as I understand. These quantities never change for an isolated system and are always proportional to each other with a factor c^2.

Staff Emeritus
Gold Member
OP was refering to total mass and total energy as I understand. These quantities never change for an isolated system and are always proportional to each other with a factor c^2.

The OP just asked about $E=mc^2$ in general. If the two sides of the equation were always simply defined as synonyms, then the equation would be vacuous. That's not what the equation expresses.

It's also not true that total mass is separately conserved. Mass isn't even additive. E.g., in Einstein's 1905 paper, the body that emitted the two light rays in opposite direction lost an amount of mass L/c2. Each light ray has zero mass. The sum of the masses has been reduced by L/c2. However, if you put the whole system, in its final state, inside a box, its inertia is the same as that of the original system, and unequal to the sum of the three masses.

What's conserved isn't mass or energy separately but mass-energy.

genericusrnme
E=mc^2 is most easily proven through an action principle imo. Check out Landau and Lifgarbagez second book, classical field theory (only the first few chapters)

haael
It's also not true that total mass is separately conserved.
It is, in all known theories. In QM for example, it comes from Noether theorem and the trivial fact that Hamiltonian commutes with itself.

It is.

E.g., in Einstein's 1905 paper, the body that emitted the two light rays in opposite direction lost an amount of mass L/c2. Each light ray has zero mass.
No, they don't. They have zero rest mass, but nonzero total mass. It is a great mistake to confuse those two different quantities.

The sum of the masses has been reduced by L/c2. However, if you put the whole system, in its final state, inside a box, its inertia is the same as that of the original system, and unequal to the sum of the three masses.
You don't need a box for the inertia to be the same. The inertia (total mass) of the system will be equal to the sum of the total masses of each particle.

What's conserved isn't mass or energy separately but mass-energy.
Total mass is conserved. Total energy is conserved and equal to the total mass with a factor c^2. Rest mass is not conserved. Kinetic energy is not conserved. Potential energy is not conserved. Rest mass (times c^2), kinetic energy and potential energy summed up together give total energy.

There are different physical quantities calles "mass" and "energy" and they are not the same despite they have similar names.

Gold Member
haael, it seems you are not aware of the convention used by almost all physicists nowadays that the term "mass" on its own is taken to mean "rest mass". The version of mass you are talking about is nowadays called "energy" (divided by c2).

Neandethal00

Utter nonsense! Pair production converts photons to electron-positron pairs. Nuclear fission or nuclear fusion convert mass to energy.

No, nuclear fission does not convert mass into energy.
It releases binding energy of a nucleus.

To me so far, E=mc2 is a conversion factor.
Conversion of Kg into Joules.

haael
haael, it seems you are not aware of the convention used by almost all physicists nowadays that the term "mass" on its own is taken to mean "rest mass". The version of mass you are talking about is nowadays called "energy" (divided by c2).
I am aware, and that is why I always give the proper adjective here (rest, kinetic, potential, total).

This jargon is very misleading, by the way. Many non-specialists in the world have hard time understanding what all this actually means.

DrStupid
It's also not true that total mass is separately conserved. Mass isn't even additive.

Classical inertial mass as defined by p=m·a is additive, the rest mass is not and both are conserved in isolated systems. Conversion of mass into energy or vice versa would violate this conservation as well as the conservation of energy.

haael
Classical inertial mass as defined by p=m·a is additive, the rest mass is not and both are conserved in isolated systems. Conversion of mass into energy or vice versa would violate this conservation as well as the conservation of energy.
Rest mass is not conserved. Consider an electron and a position annihilating into two photons. Before the annihilation, the rest mass is two times rest mass of an electron. After the annihilation, the rest mass of the system is zero.

However, the total mass of the system is conserved and is the same before and after the annihilation. Total mass of the two photons (m = hv / c^2) will be exactly the total mass of the two electrons (m = m0 / √(1 - v^2/c^2)).

Also, the real measurement of inertia is total mass, not rest mass. This is one of the explanations why any particle can not be boosted to superluminal speed - the more you try, the greater the inertia.

Rest mass is not conserved. Any particular form of energy (rest, potential, kinetic) is not conserved. Only the total mass is conserved, which is also equivalent to total energy conservation.

I am amazed how poorly this topic is understood.

DrStupid
Rest mass is not conserved. Consider an electron and a position annihilating into two photons. Before the annihilation, the rest mass is two times rest mass of an electron. After the annihilation, the rest mass of the system is zero.

The rest mass of the system is not zero. It is unchanged during the annihilation.

However, the total mass of the system is conserved and is the same before and after the annihilation.

What is your definition for the "total mass"?

haael
What is your definition for the "total mass"?
Roughly speaking: rest mass, plus kinetic mass, plus potential mass. Equivalent definition: rest energy, plus kinetic energy, plus potential energy, divided by c^2.

A free particle is characterized by 4 independent quantities: its total mass, and momentum. To convert from momentum to mass units, you have to divide it by c. These 4 numbers form a Lorentz vector called 4-momentum. Length of the 4-momentum is the rest mass, a Lorentz scalar.

The rest mass (scalar) is the same in any inertial reference frame. The total mass (and momentum) can change with reference frames. In a rest frame momentum of a particle is zero and the total mass equals the rest mass. In a moving frame momentum is nonzero and total mass increases by a factor "1 / √(1 - v^2/c^2)". This also has a beautiful geometrical explanation, as these are 4-momentum space and time coordinates in the new frame, respectively.

When you not care about Lorentz covariance, you can subtract total mass and rest mass and the result will be kinetic mass. When you multiply it by c^2, you get kinetic energy. In a small speed limit, it reduces to "m v^2 / 2". This is another way of looking at 4-momentum. It is not Lorentz-covariant, but it is preferred by some, possibly because of similarity to normal Newtonian physics.

To sum things up.
4D view:
- 4-momentum, a Lorentz vector.
- Rest mass, the length of the 4-momentum, a Lorentz scalar.

3D view:
- Momentum, space-like part of 4-momentum. It is a spatial vector.
- Total mass, time-like part of 4-momentum. It varies with speed.
- Rest mass, the same as in 4D view.
- Kinetic mass. It varies with speed. It is proportional to kinetic energy. When added to the rest mass, it gives the total mass.

Now, what it means to be conserved? Imagine a 4D spacetime with some process happening in it. Suppose we have an inertial frame. Take one moment (a space-like surface). Compute some quantity. Take some other moment. Is the quantity the same?

With this definition, 4-momentum is conserved. Total mass and momentum are also conserved, as they are components of the 4-momentu. No other quantity defined above is conserved.

Note that conservation is a different thing from Lorentz covariance. If we take some different inertial frame, momentum and total mass will be different. But they will still be conserved, that means they sum on each moment of the same reference frame will be the same.

Conservation of a quantity is its immunity to time translations. Lorentz invariance is about spacetime rotations.

This is extremely important to understand, as a common mistake is comparing total mass values from different inertial frames on different moments and "proving" its non-conservation. Namely, the rest mass is equal to the total mass in some reference frame. Some people first consider the time evolution (with some reference frame defining the time direction), then unconsiously move to the rest frame of the particle. This is not right.

To sum things up, once again:
- 4-momentum is a Lorentz 4-vector. It changes covariantly with boosts and rotations.
- Rest mass is a Lorentz scalar. It does not change with boosts and rotations. This is pure mathematics.
- Total mass is a time component of the 4-momentum.

- 4-momentum is conserved, that means for any closed system it does not change with time translations (and space-like translations too, for that matter).
- Rest mass is not conserved. It can change with time, as we can see with annihilation of two electrons.
- Total mass is conserved, since it is a time component of the 4-momentum.

I wanted to write about potential energy too, but this is too much for today. I hope I was clear enough.

DrStupid
Roughly speaking: rest mass, plus kinetic mass, plus potential mass. Equivalent definition: rest energy, plus kinetic energy, plus potential energy, divided by c^2.

That sounds like the classical inertial mass (also known as relativistic mass) but the potential energy is a problem because the potential energy between parts of the system is already part of its rest energy and the potential energy between the system and external systems is not part of the system only.

Now, what it means to be conserved? Imagine a 4D spacetime with some process happening in it. Suppose we have an inertial frame. Take one moment (a space-like surface). Compute some quantity. Take some other moment. Is the quantity the same?

This is given for rest mass as well as for the relativistic mass.

Gold Member
Rest mass is not conserved.
I am amazed how poorly this topic is understood.
Actually the real problem here is that there's no universally agreed language for explaining this. "Conservation of (rest) mass" could be interpreted in one of two ways:
1. Conservation of the sum of the rest masses of the particles of a system. This is not true.
2. Conservation of the "invariant mass", or "system mass", or "rest mass", of the whole system. Invariant mass means E/c2 where E is the total energy of the system relative to the frame in which the total momentum is zero, or equivalently $$\frac{\sqrt{\left( \Sigma E \right)^2 - \left| \Sigma \textbf{p} \right|^2 c^2}}{c^2}$$ in any frame. This type of conservation is true.
Similarly "total mass" could mean "total of the masses" or "mass of the total".

Much of the recent disagreement in this thread has been due to different contributors interpreting the language differently. I don't think they are disagreeing over the actual physics, just the terminology.

DrStupid
Actually the real problem here is that there's no universally agreed language for explaining this. "Conservation of (rest) mass" could be interpreted in one of two ways:
1. Conservation of the sum of the rest masses of the particles of a system. This is not true.
[...]

1. This makes no sense because rest mass is not additive. The sum of the rest masses of the particles of a system is not the rest mass of the system. The rest mass of the system is the mass of the system at rest.

SMarioKingdom

This is an Youtube video made by a physicist. It is made with simple drawing and it is easily understandable.

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Neandethal00

This is an Youtube video made by a physicist. It is made with simple drawing and it is easily understandable.

Here is a dumb thought.
Say, I have a ball in my hand of mass 1 Kg.
I throw it with speed 20m/sec.
Its kinetic energy is 200J. Mass equivalent of this energy is
2.22x10-15 Kg.
Can we say mass of the ball is increased by 2.22x10-15 Kg.
In another words, mass of the moving ball is 1+2.22x10-15Kg, and it will act like a ball of mass 1+2.22x10-15Kg.
Is this the mass increase we see in special relativity caused by speed?
In such case, it is not a physical increase of mass at all.
Also, then measurements of masses of planets etc are not rest masses.

There must be a big hole in my thinking.

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DrStupid
Can we say mass of the ball is increased by 2.22x10-15 Kg.

The classical inertial mass is increased but the rest mass remains unchanged.

In another words, mass of the moving ball is 1+2.22x10-15Kg, and it will act like a ball of mass 1+2.22x10-15Kg.

Define "act like"!

Mentor
The classical inertial mass is increased but the rest mass remains unchanged.

How do you define "classical inertial mass"? If you mean the "m" in F = ma, its value (in relativistic physics) depends on whether F is parallel or perpendicular to the motion of the object, or somewhere in between (at an angle).

Gold Member
Here is a dumb thought.
Say, I have a ball in my hand of mass 1 kg.
I throw it with speed 20 m/s.
Its kinetic energy is 200 J. Mass equivalent of this energy is
2.22×10-15 kg.
Can we say mass of the ball is increased by 2.22×10-15 kg?
In another words, mass of the moving ball is 1+2.22×10-15 kg, and it will act like a ball of mass 1+2.22×10-15 kg.
Is this the mass increase we see in special relativity caused by speed?
Yes to all of the above*. However, there is more than one definition of mass in relativity, and this version ("relativistic mass") is a version most physicists don't like to use any more. They prefer to work in terms of rest mass only (so they usually call rest mass just "mass") and they use equations involving rest mass and not relativistic mass.

*To be precise, the kinetic energy calculated using relativity rather than Newtonian physics would be a tiny, tiny amount larger than 200 J.

DrStupid
How do you define "classical inertial mass"?

It was implicit defined by Newton as the "m" in p=m·v

Neandethal00
Yes to all of the above*. However, there is more than one definition of mass in relativity, and this version ("relativistic mass") is a version most physicists don't like to use any more. They prefer to work in terms of rest mass only (so they usually call rest mass just "mass") and they use equations involving rest mass and not relativistic mass.

Then we have an interpretation problem with special relativity.
Usually we say, 'one reason speed faster than light is not possible is mass of the
object will become so high it would require all energy of the universe to move'.
This statement then is not correct. The actual mass the energy would work on
is always the rest mass, increased mass is just its energy.

Define "act like"! DrStupid

For example, mass of the moon we use for gravitational pull may not be its rest mass,
but rest mass + energy from its motion.

arindamsinha
There does not seem to be any real 'exact' proof of E = mc2. This was a very good and logical approximation made by Einstein in his 1905 paper "Does the Inertia of a Body depend on its Energy-Content?"

No substantial experimental proof has yet appeared to disprove this, so we must assume this is close to correct (perhaps similar to accepting that Newtonian gravitational theory is close to correct, till we learnt better). There is the possibility that there may be additional correction factors with small influence which have not yet been discovered.

I have seen some claims that this has been proved 'exactly' later on, using 4-vectors etc. in GR, but from what I can make out, those depend ultimately on the assumption of correctness of this equation itself.

I am no real expert in relativity theory, so the above is based on my limited knowledge and understanding.

Gold Member
Then we have an interpretation problem with special relativity.
Usually we say, 'one reason speed faster than light is not possible is mass of the
object will become so high it would require all energy of the universe to move'.
This statement then is not correct. The actual mass the energy would work on
is always the rest mass, increased mass is just its energy.
You just need to argue in terms of energy instead of mass. As I hinted in the tiny footnote to my last post, the correct relativistic formula for kinetic energy isn't $\tfrac{1}{2}mv^2$, it is really$$mc^2 \left( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1 \right)$$where m is rest mass, which approximates to $\tfrac{1}{2}mv^2$ when (v/c) is very small. So an object with non-zero rest mass would have infinite kinetic energy if it could travel at the speed of light -- impossible.

DrStupid
For example, mass of the moon we use for gravitational pull may not be its rest mass, but rest mass + energy from its motion.

Gravitational mass of a moving body does not necessarily be equal to its inertial mass. The weak equivalence principle is limited to bodies at rest.

shivaniits

No, nuclear fission does not convert mass into energy.
It releases binding energy of a nucleus.

To me so far, E=mc2 is a conversion factor.
Conversion of Kg into Joules.

ok so nuclear reactions don't produce something from nothing but they however converting the electron positron pairs from neutrons that is something that resides in nucleus and the relation e=mc^2 is just a conversion factor to judge the energy of a body...!! can i say this by summing all..??