Stopping time, by speeding it up inside a bubble
Imagine I have a device that can stop time for the person who holds it (similar to Bernard's Watch). This device works in a very specific way - it creates a bubble around the user (just large enough to hold the user) in which time flows much faster than in the rest of the universe. The difference is very high but it is finite (say 100,000 times faster). This means that if the person holding the device experience times at "normal speed" the rest of the world outside the bubble appears to him to be drastically slowed down, almost (but not quite) to a standstill. In contrast, anyone outside the bubble looking in would see everything inside the bubble happening at lightning fast speed (almost instantaneously).
The border between this bubble of faster time and the rest of the universe is not infinitely thin - there is a boundary (say a few centimetres thick) where the speed of time changes gradually from one to the other.
My question is what side effects would this bubble cause? For example: any sounds from the outside would sound quieter and deeper, as the wavelengths are "stretched out" at the boundary. Similarly looking out of the bubble the outside world would appear darker and redder, as light is red-shifted. In fact with enough time dilation you would be able to see x-rays with your naked eyes. I'm also positing that waves such as light and sound would also bend at the boundary, as if they had struck a lense, so the outside world would appear distorted. Moving object such as bullets would also be deflected slightly (if they struck at an angle).
Are there any other interesting effects that could arise? What would happen if you walked up to another person, as they crossed the bubble's boundary? Is there anything cool you could do with this device?
For extra credit: would this violate any physically laws in any way e.g. conservation of energy? Show your working!
This post was sourced from https://worldbuilding.stackexchange.com/q/57925. It is licensed under CC BY-SA 3.0.
1 answer
Well, let's see how such a field would work. The Hamiltonian equations are $$\frac{\mathrm d x_k}{\mathrm d t} = \frac{\partial H}{\partial p_k} \quad \frac{\mathrm d p_k}{\mathrm d t} = -\frac{\partial H}{\partial x_k}$$ Now we want the field to affect the speed of things happening, so a natural assumption would be that the field just acts as factor of the Hamiltonian: $$H(x,p) = \exp(f(x,t)) H_0(x,p)$$ Here the exponential function mainly is there to make time go normally when the field is zero. It however also makes sure time cannot go reverse.
Now for a constant non-zero field, you'd just get a rate of temporal change proportional to $\exp(f)$, so if the field is positive, things are going faster, as intended.
However what happens in the "bubble wall", assuming a static field? Well, here we have to use the product rule: \begin{aligned} \frac{\mathrm d x_k}{\mathrm d t} &= \exp{f(x)}\frac{\partial H_0}{\partial p_k}\\ \frac{\mathrm d p_k}{\mathrm d t} &= -\exp(f(x))\frac{\partial H_0}{\partial x_k} - f'(x)\exp(f(x))H_0(x,p) \end{aligned} Note the extra term on the second equation. This is an extra force proportional to the derivative of the field and the total Hamiltonian (which more or less gives the total energy). Assuming that energy is positive, this means that this acts like an extra repulsive force. Note that this repulsive force is in addition to the slowing down due to the local factor, and its strength depends on how rapidly the field grows. Note that since the kinetic energy is proportional to the mass, this leads also to a force term proportional to mass, similar to gravitation. Any potential energy would, however, give rise to an extra force that is not mass dependent.
Let's look at the energy, now also with time-dependent field: $$\frac{\mathrm dE}{dt} = \frac{\partial H}{\partial t} = \exp(f(x,t))\frac{\partial H}{\partial t} + \frac{\partial f}{\partial t}\exp(f(x,t))H_0(x,p)$$ So energy change only happens with field change, and proportional to it; that can well be explained as energy going into the field (of which I only included the effect). As long as the field remains constant, energy is conserved.
In effect, the outside world would look colder than would be expected from the slowdown, as the additional force would draw away more energy from incoming particles;on the other hand, outgoing particles would get an extra boost, so to the outside world, you'd be hotter than expected from the speedup.
Indeed, with a sufficiently small border zone (and corresponding rapid onset of the field) the border might even act like a wall for all normal-speed particles coming from outside.
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