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For example, if the equation for energy were:

$$E = mc^{2.713397972993}$$

clearly $2.713397972993$ would be a universal constant.

And in the Einstein field equation:

$$R_{\mu \nu} - \tfrac{1}{2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}$$

Why would $Λ$ be considered a constant but $1/2$, $8$, and $4$ are not?

Or put another way, why do squared, cubed, $2$, $1/2$, etc. appear in so many fundamental formulas of the universe?

And yes, I know the end result is a derivative of other universal constants, but it still seems signficant that it's $c^2$ and not $c^{11}$ and $8πG$ and not $25πG$.

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closed as primarily opinion-based by Buzz, Jon Custer, Norbert Schuch, GiorgioP, Emilio Pisanty Mar 28 at 13:35

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ $c$ is dimensionful. The dimensions on LHS and RHS won't match for any power of $c$. Also, how do you define the dimensions of $c^{2.7133...}$? $\endgroup$ – Avantgarde Mar 28 at 2:09
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    $\begingroup$ You really need to study physics a LOT more before you decide what a universal constant is. $\endgroup$ – David White Mar 28 at 2:11
  • $\begingroup$ It could be $c^{80083}$, the point is the number is significant. $\endgroup$ – Animism Mar 28 at 2:12
  • $\begingroup$ It can only be 2 in order for the dimensions to match LHS & RHS. Similar deal for most other formulae. What we call "constants" in Physics are typically irrational numbers, and in truth they are not actually constant (e.g. c) $\endgroup$ – Paul Childs Mar 28 at 2:15
  • $\begingroup$ @Avantgarde OP is not specific about dimensions, for example $l=r\theta$ but why not $l=r\theta^2$. $\endgroup$ – PranshuKhandal Mar 28 at 5:05
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2, 1/2, $\pi$, etc. are indeed universal constants, but they are universal mathematical constants rather than universal physical constants. They are either known exactly or they can be calculated to an arbitrary degree of precision purely mathematically.

Another class of constants are the constants like $c$ or $G$. These are universal constants that have dimensions. These seem more likely to be physical constants, but it turns out that you can always choose a system of units such that any of these constants is known exactly. Indeed, in the current SI system $c$ is already such an exactly defined constant, and soon $h$ and several others will join it. These constants therefore do not tell us anything about physics beyond our choice of units.

A final set of constants are the dimensionless universal constants, the most famous being the fine structure constant, $\alpha$. These constants are dimentionless, so they are independent of our system of units, and they cannot be determined mathematically but only experimentally. These are the constants that really are the important physical universal constants.

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  • $\begingroup$ How do you know the fine structure constant is not a mathematical constant like $\pi$? $\endgroup$ – Ryan Thorngren Apr 22 at 21:03
  • $\begingroup$ Because there is no known mathematical formula to calculate it to arbitrary precision. $\endgroup$ – Dale Apr 22 at 23:15
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I'll leave aside the whole discussion on whether $c$ is a fundamental constant or a unit conversion factor, and take your question at face value. In your particular example, $E=mc^2$ the power of $c$ can't be anything other than 2, simply due to dimensional analysis. If you want to introduce an exponent other than two, you would need to introduce something extra with units of velocity, $v$. Then you could write the formula, $$E=mc^2\left(\frac{c}{v}\right)^{0.713397972993},$$ the point being you need to introduce a 'universal constant' you didn't anticipate, the dimensionless ratio $v/c$. But once you've done this you could indeed think of this anomalous dimension 0.713397972993, as a physical constant.

The example with $E=mc^2$ is a little artificial but this general picture really does show up in physics. In quantum field theory you often see formulas modified by anomalous dimensions which are related to physical coupling constants like the fine structure constant. (The analogue to $v$ in this picture would be some kind of renormalization scale).

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