What would be the equation for kinetic energy if $γ=\frac{v^2}{c^2}+1$ instead of $γ=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$?

Question

I was thinking of a universe where the equation for the Lorentz factor $$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ was replaced with the equation $$\gamma=\frac{v^2}{c^2}+1$$ so in this hypothetical universe there would be the effect of length contraction and time dilation and the inertia of an object would appear to increase as it moves very fast, so as something would move very quickly, it would appear to be harder to accelerate, but there would be no cosmic speed limit. So in this universe $c$ would simply be a constant instead of an actual speed limit.

What would be the new equation for kinetic energy for a universe with this equation? Also, what would be the new equation for rest mass energy in this hypothetical universe?

Answer 1

Let's start with the energy-momentum equation: $$E^2=p^2c^2+(m_0c^2)^2\tag{1}$$ This can be derived according to the Minkowski metric. This works because the inner product of the four-momentum, $\langle\mathbf{P},\mathbf{P}\rangle$, is equal to $|\mathbf{P}|^2=-(m_0c)^2$. We can also use $$\langle\mathbf{P},\mathbf{P}\rangle=P^\alpha\eta_{\alpha\beta}P^\beta=-\left(\frac{E}{c}\right)^2+p^2\tag{2}$$ where $\eta_{\alpha\beta}$ is the Minkowski metric. Setting these two expressions equal yields $(1)$. We can then use this to derive an expression for $\gamma$.

Now let's do things in reverse, with your requirements. First, let us rewrite your $\gamma$ as $$\gamma=\frac{c^2+v^2}{c^2}$$ Putting this into the expression $$E=\gamma m_0c^2$$ We find $$\frac{E}{c^2+v^2}=m_0$$ We then have $$E^2=p^2v^2+(m_0c^2)^2\tag{3}$$ Note that we have $p^2v^2$, instead of $p^2c^2$. We now have $$-(m_0c^2)^2=-\left(\frac{E}{c}\right)^2+\frac{p^2v^2}{c^2}=\langle\mathbf{P},\mathbf{P}\rangle$$ This implies that you have a metric that is nothing like the Minkowski metric, and you have spacetime that is nothing like Minkowski spacetime. The final term now includes a dependence on $v$. Now you have a problem, because special relativity needs Minkowski spacetime to work. The postulates of special relativity, especially those concerning invariance, most likely will not hold.

The big problem with this - all of this - is that you haven't started from first principles. Instead of using some logic to make a derivation, you've done things the other way around, starting from a result you want and trying to work backwards. You're then left with results that might be described as disastrous.

This is an easy trap to fall into. You would think that changing one tiny thing about a universe wouldn't cause too many problems, but it can. Each equation, each law, each postulate that makes up our universe is finely woven together with every one to form a self-consistent framework that describes how things work. It's like making a jigsaw puzzle, where each piece is a different law of nature. You can change the shape of one piece, and change the shape of one of the neighboring pieces to compensate. But unless you modify all of the pieces that touch the modified piece, the puzzle won't be self-consistent.

Answer 2

See Nothing But Relativity and similar papers: you can see from from first principles that the Lorentz factor is what it is, not something than can be arbitrarily changed.

So the underlying principles must have changed. These are the symmetry of space and time. The idea that if Alice sees Bob moving at some velocity, then Bob must see Alice move in the equal but opposite velocity! That space is the same everywhere and in both directions (of a line) and in every direction (any line in 3D you choose). That the rules do not change with time.

Now watch the first few videos of The Mechanical Universe which is available for free online. You'll see that kenetic energy is simply the total amount of energy you put in when bringing an object up to speed. And you would need that much to stop it, or would get that much out if you could extract it to use for some other purpose.

Except that you already broke concervation of energy as noted above. So as a concept it just doesn't have any use in your universe. The amount of energy needed to put in to bring the object up to speed will vary over time, depend on direction, or depend on location; and how much you can get out, even just by reversing the direction of thrust, might be totally unrelated to the first value.

So the question has no answer. Kenetic energy (as we understand it) will not be a thing. See also this post.