# Hypothetical mass-energy equivalence

2 questions:

1. Hypothetically, if in an imaginary universe, the speed of light squared were 90% or 110% of what is is in our universe...would the math require another constant such as 1.1 or .9 next to it to make the math work (because of momentum requirements)?

2. Under a slightly different scenario, what if the $c$ observed were the same as in our universe, but because of some property of mass (different in the alternate universe), the math requires a constant of 1.1 or .9 for the math to work out. Is this a possibility in math, considering momentum and conservation of energy in this imaginary universe?

all physically meaningful measurements are dimensionless: wavelengths, frequencies, masses, etc. are all measured in ratios of a standard quantity with the same units. Masses can be all expressed in electron masses, wavelengths can be expressed in comparison to the wavelength of a certain sodium absorption line.

If speed of light were up by a factor of 1.01 or 0.99, keeping all the rest of physics constant, we need to study how the change in a single constant propagates among all the other predictions before we can assess that the change is itself physically observable, or if it is fixed by other physical constants

The answers are "not really, and no".

Special relativity is based on the two postulates that (1) the laws of physics are the same in all inertial frames of reference, and (2) that the speed of light in vacuum has the same value $c$ in all inertial frames of reference.

The reason we express $c$ as $\approx 300,000$ km/s is that we chose the units kilometre and second, but we could just as well use units of length and time that make $c=1$. So you cannot just change $c$ on its own, since it would just rescale distance and time. Normally when physicists consider different values of the fundamental constants they look at dimensionless constants, that is values that do not have any units and hence cannot be rescaled by changing your system of units. So changing $c$ requires changing a few of the other constants, changing a fair bit of physics. However, let's ignore that part to get to the question - just assume we mess with the constants in the right way.

Does the fact that light travels at $c$ do any work here? Apparently not: the only thing needed is that there is some signalling speed that is invariant in inertial reference frames. The causality goes the other way around: since photons are mass-less (or, alternatively, Maxwell's equations are Lorenz-invariant) light have to travel at the invariant speed. Relativity would still hold if space had a refractive index $n>1$ slowing light down.

The first question is whether in a universe with a different value of the invariant speed $E=mc^2$ would have to change. The quick answer is no: in this universe $E=mc'^2$ where $c'$ is the changed speed (if you want to express this formula in terms of normal $c$ you will need to add a factor in front of it, but now you are expressing one the observed constant in terms of one from another universe - it doesn't make much sense, and you will have to do it everywhere $c'$ shows up in your equations).

The reason is that the derivation (variants are found in all relativity textbooks) only makes use of how momentum and mass transforms based on the invariant speed, not what value it has. It can just be treated as a symbol: there is no link to actual light or a particular value.

(Strictly speaking, Einstein's original derivation was a lot about emitting photons and seems to make light much more important to the result than it is.)

The second question is whether we could end up with an equation like $E=2mc^2$. The answer is no for the same reason. The math will not work out. In particular, consider the energy-momentum formula $E^2=(mc^2)^2+(pc)^2$ - if you want this to hold and $E=2mc^2$ then you get an imaginary momentum. So unless you want to postulate a universe with a really different physics you are stuck with $E=mc^2$.

• Photons' being massless isn't equivalent to Maxwell's equations being Lorentz-invariant. There exists a generalization of Maxwell's equations with massive photons that's still Lorentz-invariant. It describes E&M inside of superconductors. – tparker Aug 16 '18 at 22:09

The speed of light is a distance that light covers during 1 second. This distance is 1 light-second or exactly 299,792,458 meters by the definition of meter. This speed is not some measured physical constant that could be different in a different frame or in a different universe. Instead, the speed of light is a predefined number that is always the same. No matter where you are, inside or outside a black hole or in a different universe, your local speed of light always is 1 light-second per second.

On your second scenario, there is no property of mass that could change the relation. Energy and mass are not in a "relation", mass simply is the local energy (energy that remains in the rest frame of the object). For example, if you take a weightless box with ideal mirror walls keeping light inside, then the mass of this box (in natural units where $c=1$) equals the energy of the light. We add $c^2$ (a predefined number from the definition of meter) to the formula only to convert Joules to kilograms. In natural units the formula simply is $E=m$.

1) No, they would not need to use a constant factor to make the math work out. Assuming the properties of matter hold true, then their version of the speed of light would simply be their limiting speed of information transfer. Their equations would look exactly the same as ours, except that they would use their measured value for the speed of light instead of ours (because why would they use or even know about ours?) They would still say that $E=mc^2$

2) This is obviously a possibility of math. Unfortunately, the answer to your question about this scenario is a trivial one. If we establish that the physicists of an alternate universe measure the speed of light to be the same as ours but that there is some property of matter that makes their mass-energy equivalence come out to $E=kmc^2$ for some constant, $k$, then the math would necessarily allow for that.

We don't say $E=mc^2$ because of some purely mathematical derivation that makes it impossible for the relation to be anything else. We use that equation because that is best supported by the scientific evidence. Whatever relation is best supported by evidence is also possible within mathematics. A different relation might mean that there are changes, large or small, to other theories and principles within physics, but the math has no issue with that.

So your second scenario is trivial and works out because you told it to. Whatever physicists experimentally determine in that universe is what they determine. You told us to set up a hypothetical in which $E=0.9mc^2$ and asked if the math worked out. Well, if it were hypothetically the case that $E=0.9mc^2$, then the math would have to work out, otherwise we wouldn't be adhering to the hypothetical.