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mass-energy equivalence
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Definition/Summary
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An attempt to solve common confusion and misconceptions regarding mass and energy equivalence. Somewhat FAQ related post.
Energy (measured in joules, or kg m˛/s˛) has dimensions of mass times distance˛ per time˛, which is the same as mass times speed˛.
This page is about rest-energy, the energy that matter has independently of its speed its temperature its position or any other variable.
The rest-energy of matter is its mass (rest-mass) times the speed of light squared. |
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Equations
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| [tex] E = m_0 c^2 [/tex] |
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Recent forum threads on mass-energy equivalence
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Breakdown
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Physics
> Relativity
>> Miscellaneous
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Extended explanation
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The equation [itex] E = m_0 c^2 [/itex] is perhaps the best know formula in all physics, but also one of the most confusing. There are many misconceptions and this post will try to clarify the significance of this formula.
Let us first see what this formula contains. First, we have the rest mass, [itex] m_0 [/itex], which is the mass measured in the body's own rest frame. And [itex] c [/itex] is the speed of light in vacuum. What this formula is mathematically is a straight line equation, relating the energy to the rest mass, with [itex] c^2 [/itex] as constant of proportionality. The main difference of mathematics and physics is that physics deals with units - the constant of proportionality is there to fix the units on the right hand side with the one on the left hand side.
In SI-units, this conversion factor is measured in units of [m/s], and energy is measured in [J] and mass in [kg]. Thus we have this unit-equation: [Energy = Work = Force x Path = Mass x acceleration x path = kg x (m/s2) x m = kg x (m/s)2 = Mass x Velocity2 = J]
The physics should not depend on what units we assign to the speed of light, energy and mass. Units are our gauge, so we can compare results with each other. Physicists in Sweden must be able to communicate with American physicists. American physicists tend to use the system with inches and so on, and Swedes use SI units. But physics is totally independent of this. The results must agree when compensating for the different units used. There is nothing special about the choice of units we choose!* Imagine an extra terrestrial intelligent race had a totally different way to measure mass and length, etc.
There is no reason for why we can't let the speed of light be unit-less (we measure everything else in terms of light speed). So let [itex]c = 1[/itex]. Then our units for mass is measured in energy / c2 = energy. The equation becomes [itex]E = m_0[/itex]. This is done all the time in, for example, elementary particle physics; energy, mass, and momentum are measured in [electron Volt].**
Let us now consider a photon, light, which has no rest mass. Its energy is given by [itex] E = \omega \hbar [/itex]. [itex] \omega [/itex] is the angular frequency of the electric- and magnetic field associated by the photon. [itex] \hbar [/itex] has in SI units [J s] which is just a conversion factor, as [itex]c[/itex] was for the energy of a massive particle. The actual physics depends on [itex] \omega[/itex] and [itex]m_0[/itex], not the unit-gauge factors. 
We can chose to work in units where the angular frequency is measured in terms of [itex]\hbar[/itex] instead of turns per second [s-1 ].*** Then the equation becomes [itex]E = \omega[/itex].
Thus, we have demonstrated that energy can be manifested in different ways; rest mass, angular frequency of fields. 
Mass is one kind of energy, and so is angular frequency of EM fields.
Theorem:
Conservation of energy is a consequence of the symmetries of physical laws under time transformations - time invariance ****.
So with this, we can summarize what we have discussed so far into:
Energy can not be created or be destroyed. It can only be transferred into different forms.
Let us draw our attention to a process as an example:
[tex] e^+ + e^- \rightarrow \gamma \gamma ,[/tex]
where [itex] \gamma [/itex] represents a photon.
The common description "mass has became energy" is a bit misleading. What has happened is that energy has transferred form, from mass to angular frequency. The energy is conserved, in accordance with the theorem stated above. Photons are not 'pure' energy.
Another thing to take into mind when it comes to the property of photons is gravitational effects. It might be believed that since gravity is due to mass, and photons are massless, photons does not follow gravity.
Well, true, in some sense. But that is gravity according to Newton, not according to gravity of Einstein where gravity is a space-time effect which invokes energy, not mass.
Examples are gravitational lensing of light and gravitational red shift. Also a container filled with photon gas will induce a gravitational field.
*) compare with phase choice of fields which is another kind gauge fixing, physics should independent of the choice of phase and this leads to particle interactions in field theories. 
**) One might think that those particle physicists are one example extra terrestrial intelligence  
***) This choice is also done frequently by particle physicists.
****) See Noether's theorem, e.g. on wikipedia: http://en.wikipedia.org/wiki/Noether...tion_of_energy
mass-distance equivalence:
We could measure almost everything in terms of time, if we wished.
For example, we do measure large distances using the the light-year, a unit equal to the year (a time) times the speed of light.
Alternatively, we could measure everything in terms of distance, for example measuring time by using a "light-metre", which would be a metre divided by the speed of light, the time taken for light to travel one metre.
Either of these methods gives an equivalence of time and distance, using the conversion factor [itex]c[/itex].
Similarly, there is an equivalence of mass and distance, using the conversion factor [itex]G/c^2[/itex] (where [itex]G[/itex] is the universal gravitational constant), giving the mass-equivalent radius of a body:
[tex]M\ =\ \frac{Gm}{c^2}[/tex]
used in such formulas as [itex]\text{Schwarzschild radius}\ =\ 2M[/itex]. |
Commentary
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