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I am aware that energy mass are inter-convertible using the famous $$E=mc^2$$

But why is it that energy and mass are basically the same thing that takes different forms?

I am looking for a theoretical answer rather than answer based on formula. I am also changing matter to mass as pointed out in the comments.

Charlie
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Danny LeBeau
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    Energy and mass. Not matter. – Valter Moretti May 23 '21 at 13:55
  • Have a look at this relevant answer of mine to a similar question https://physics.stackexchange.com/questions/525047/what-happens-to-matter-when-it-is-converted-into-energy/525122#525122 – anna v May 23 '21 at 14:02

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The $m$ in the famous $E=mc^2$ is “mass”, not “matter”. Mass and energy are properties of matter, but they are not the only properties. Matter also has other properties like spin and charge.

Regarding mass and energy. Although super-famous, the formula $ E = mc^2$ is a simplification of a more general formula: $E^2/c^2-p^2=m^2 c^2$. The famous formula only applies for the special case of $p=0$. In general mass and energy are different, but they are related to each other and to momentum by the more general formula.

Specifically, mass, energy, and momentum are all parts of the relativistic four-momentum. Energy is the component of four-momentum in the time direction and momentum is the component of four-momentum in the space direction. Then mass is the magnitude of the four-momentum.

Dale
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  • More than a simplification, I would call it a particular case. – GiorgioP-DoomsdayClockIsAt-90 May 23 '21 at 14:03
  • Sure. I said “special case” instead of “particular case”, but I think that is equivalent. – Dale May 23 '21 at 14:12
  • When we consider mass to vary by a factor $\gamma$, $E=mc^2$ is the general formula for total energy. – Physiker May 23 '21 at 14:30
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    The idea of relativistic mass is somewhat outdated though, this gets asked a lot on the site. – Charlie May 23 '21 at 15:18
  • Can you explain how "In general mass and energy are different" ? – Danny LeBeau May 23 '21 at 15:22
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    @LoneAcademic the majority of professional scientists, including myself, do not consider mass to vary by $\gamma$. I am not going to include relativistic mass in my answer, which currently captures the broad scientific consensus. – Dale May 23 '21 at 16:12
  • @Danny LeBeau I have added a paragraph. Hopefully it helps, but you may need to ask a completely separate question – Dale May 23 '21 at 16:41
  • @Dale I would also add that we don't need to vary the mass by $\gamma$ to perform the derivation, it's just what some old derivations had done. To me it seems that the more fundamental equation for momentum inherently includes $\gamma$, e.g. $p=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}v$ (maybe this is an oversimplification?) – michael b May 24 '21 at 17:04
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This is because of SR (in the following $\eta_{\mu \nu}=diag(-c^2,1,1,1)$):

We have that for the four-velocity $u$ of a massive particle it holds:

$$u^\mu u_{\mu}=-c^2$$

Since the 4-momentum is $p=mu$ we have:

$$p^\mu p_\mu =(m u^\mu)( mu_\mu)=-m^2c^2$$

But since $p=(E/c^2, \vec{p})$:

$$p^\mu p_\mu =-c^2\frac{E^2}{c^4}+|\vec{p}|^2=-m^2c^2$$

I.e.:

$$E^2-|\vec{p}|^2c^2=m^2c^4$$

That reduces to $E=mc^2$ in the rest frame of the particle (where the three-momentum $\vec{p}$ is null).

Andrea Di Pinto
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One possible way to look at it is that we tend (anthropomorphism) to regard processes as being fundamentally different from objects and to class physical phenomena into these two categories (objects, processes), as if these two categories were inherently mutually exclusive. Quantum mechanics teaches us that the particle/wave duality is erroneous; electrons, photons have both characteristics of a particle and a wave.

Another example of physical phenomena which appear at first to be objects but prove to behave in a more dynamic fashion on closer look is given by the example of protons or neutrons. Most of the mass of the proton (or neutrons) does not result from the sum of the masses of the quarks but from the average kinetic energy of these particles. So basically, most of the mass of the p/n is kinetic energy; therefore, most of the mass of nuclei is kinetic energy, and by extension, most of the mass of atoms (or matter more generally) is kinetic energy. https://www.forbes.com/sites/startswithabang/2016/08/03/where-does-the-mass-of-a-proton-come-from/?sh=791273ea2e1d

Mass, therefore, is essentially energy; stated otherwise, mass is just an anthropomorphism, a macroscopic designation, just a word for a more complex, dynamic phenomenon. To put it another way, one couldn't convert mass into energy and conversely, if at some level these two facets of matter were not one and the same thing.

Serge Hulne
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