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To clarify, I know very little about physics and don't pretend to have any insight whatsoever into relativity beyond what has entered the popular imagination; my knowledge is more or less at the level only of someone who has taken vector calculus and beginning differential geometry.

Nevertheless, it has struck me that the warping of space by mass (see picture) is "like" curvature $\kappa$ and that lightspeed $c$ is "like" velocity $v$ in the formula for the normal component of acceleration $\kappa v^2$; in other words that there is a very general similarity in form between the expressions $m c^2$ and $\kappa v^2$.

enter image description here

Also, it seems to me that if one imagines electrons orbiting an atom (with a very high curvature $\kappa$ for the orbit since $\kappa = 1/R$ for an approximate circle), that the correspondingly high normal component to acceleration would equate to a lot of energy, for example if the electrons suddenly were released from their orbits. Is there any correct intuition here at all? Again please remember if answering that this is purely speculative - but the site is here to ask, so I'm asking. Please don't be too hard on me if this is preposterously silly for anyone that really understands relativity!

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    $\begingroup$ I don't see any. $E=mc^2$ is not even a full equation. On a side note, Energy is usually written with a capital E to distinguish it from the electron and Euler's number. The correlation of these unrelated quantities sounds a bit like magical thinking to me. Maybe somebody else has an idea... $\endgroup$
    – CuriousOne
    Dec 30, 2014 at 21:04
  • $\begingroup$ Hi, maybe you could read : en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation ? $\endgroup$
    – TZDZ
    Dec 30, 2014 at 21:15

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Yes, of course there's a connection between the two, but you shouldn't read too much into it, since there is a great number of mathematical relations of the form $x=y\dfrac{z^2}2$ in physics. Thus, for instance, we have $E_k=\displaystyle\int p(v)~dv=\int mv~dv=m\dfrac{v^2}2$ , denoting the relation between kinetic energy and momentum, or

$E_C=\displaystyle\int Q(U)~dU=\int CU~dU=C\dfrac{U^2}2\quad$ and $\quad E_L=\displaystyle\int LI~dI=L\dfrac{I^2}2~,~$ for the electric energy

stored on a capacitor (or inductor), expressed in terms of its capacitance and voltage (or inductance and current). There are countless such parallelisms and analogies between the various branches of physics, such as mechanics and electromagnetics, etc. We can even go further, into the realm of geometry, and write $A=\displaystyle\int C(r)~dr=\int2\pi r~dr=2\pi\dfrac{r^2}2=\pi r^2~,~$ linking the area or surface of the disk or circle to its radius and circumference, etc.

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    $\begingroup$ if there's a connection between the two, then what IS the connection? I feel like they're pretty different but maybe there's a way of seeing it that relates them. $\endgroup$
    – doublefelix
    Dec 31, 2014 at 8:43
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    $\begingroup$ The main connection between the two pairs, $(E,c)$ and $(a_n,v)$ is the way in which the two elements of each relate to one another. Basically, both $E(c)$ and $a_n(v)$ are solutions to differential equations of the form $y'(x)=A\cdot x$. But such differential equations are ubiquitous in nature. Furthermore, c and v are both speeds of something. $\endgroup$
    – user68746
    Dec 31, 2014 at 9:00
  • $\begingroup$ Just to nipick a little, I would argue that your example of the area of a circle should be written $\iint_{x^{2}+y^{2}\leq r}1.dxdy$ then calculate the integral using polar coordinates $\int^{2 \pi}_{0}\int_{0}^{r}r dr d\theta = \int^{r}_{0}2 \pi r dr=\pi r^{2}$ $\endgroup$
    – Autolatry
    Jan 30, 2015 at 11:22

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