If $E= mc^2$ then c = $\pm \sqrt{E/m}$. What is the real world meaning for a negative speed of light ? Anything traveling faster than $-300k m/s$ is travelling faster than the speed of light ? What have I got wrong ?
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2$\begingroup$ Say I have three apples ($n=3$), and then I define $x = n^2 = 9$. This process does not imply that $-3$ is a valid number of apples. $\endgroup$– tom10Commented Mar 10, 2021 at 16:28
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$\begingroup$ The language of physics is mathematics, but physics is NOT mathematics. That negative value for speed is a mathematical artifact that does not exist in the real world. $\endgroup$– David WhiteCommented Mar 10, 2021 at 17:18
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1$\begingroup$ @tom10: or for that matter, I have n=3 apples. Therefore my number of apples satisfies $n^2-7n+12=0$. But $n=4$ satisfies this equation. Therefore I have four apples. $\endgroup$– WillOCommented Mar 10, 2021 at 17:50
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1$\begingroup$ @DavidWhite I agree more broadly but here, it is not even a mathematical artifact. Nobody promised that the domain of $c$ is $\mathbb{R}$. $\endgroup$– user87745Commented Mar 10, 2021 at 18:43
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1$\begingroup$ The speed of light is a shade under a thousand times faster than 300 km/s $\endgroup$– notovnyCommented Mar 10, 2021 at 19:43
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3 Answers
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Speed is a scalar quantity - it only has magnitude, and no direction. The speed of light does not reference directionality in any way - it's simply a distance per time, but we don't care in what direction that distance is. The concept of a negative speed doesn't make sense, as that tries to layer directionality onto a scalar quantity. If you drive your car backwards, your speed is still positive. It is not possible to have negative speed. This would imply that you traveled negative distance, which is also not possible - the distance between two objects cannot be less than zero.
As another analogy, the area of a square is simply the length of a side squared. But that does not imply that you can have a square with negative side length.
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1$\begingroup$ Scalars can be negative. Maybe it would be better to say "speed is the magnitude of the velocity vector, and is therefore non-negative." $\endgroup$ Commented Mar 12, 2021 at 22:09
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Suppose I have two bodies of equal mass $m$, so the gravitational attraction between them is $f = G m^2/r^2$. Does it follow from this that both masses could be negative? Or could the distance between them be negative? No, those ideas do not logically follow from $f = G m^2 / r^2$. Similarly, there is no logical implication from $E = m c^2$ to the idea that $c < 0$. One can logically infer various things, such as for example that if $c$ were negative then the energy would still be positive. But this does not allow us to deduce that $c$ can be negative.
The skill of telling the difference between physics and maths is an important part of physics, and it is not always straightforward; there can be a bit of an art to it. Sometimes the various alternative solutions to a given equation all represent physical possibilities, sometimes they do not. The way to tell which is the case is to bring in further information. In the present example we already know by other means that $c$ is positive. It follows that $(-|(E/m)^{1/2}|)$ is not a quantity that ever equals the speed of light.
answered Mar 12, 2021 at 22:50
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There is a simpler and more direct answer.
E = mc^2 is a special case, of the more general relativistic energy-momentum equation. This is specifically the case of zero inertia in a centre-of-momentum frame. That is, if the object were to move in a direction, it would define a reference frame by making that direction positive.
QED.
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$\begingroup$ If you disagree, by all means comment why $\endgroup$ Commented Mar 16, 2021 at 15:52