Would time dilation alter apparent mass observed from outside time dilation zone?

Question

A familiar scenario in scifi: a bubble of space is isolated artificially from the outside world by means of time dilation: time inside the bubble moves more slowly than time outside it. Assume that the boundary is blurred. Time dilation gradually increases as you approach the center. This is similar to time dilation caused by gravity. Also assume time at the center does not stop but is simply the slowest.

There is an object which lies inside the bubble and we attempt to move it (edit: The bubble is small enough so that someone from the outside may attempt to move it using a tether or a rod). From our perspective, our action may appear normal. From the object's perspective we move it faster than we intend to. That raises a question: by moving that object, do we experience more inertia as if the object was more massive? Does time dilation make us perceive that the object in the bubble is more massive than its classical mass?

A second effect is light crossing the bubble: light hitting the bubble's boundary at an incident (non perpendicular) angle, will have the wave nearer to the bubble's center move more slowly. Will that cause light to diffract? The reason light diffracts when crossing a prism is the slower light speed within the prism, and that's what defines the refractive index. So does the bubble do the same by means of time dilation? Unlike a prism, the bubble has no sharp boundaries. Light entering at an angle should "curve" rather than diffract abruptly.

I expect light emanating/reflecting from the object to red-shift to a viewer outside the bubble, but this is not expected from light merely crossing the bubble's space. Will the visual effects resemble gravitational lensing by massive objects?

Answer 1

This question is all about relativity. The second part that asks about the effects on light are essentially about gravitational lensing.

In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In General Relativity the speed of light depends on the gravitational potential (aka the metric) and this bending can be viewed as a consequence of the light traveling along a gradient in light speed.

There should be the equivalent of a gravitational redshift for light coming out of the time dilation bubble. This is due to different clock rates inside and outside the bubble, or the different rates of the passing of time.

The changing rates of clocks allowed Einstein to conclude that light waves change frequency as they move, and the frequency/energy relationship for photons allowed him to see that this was best interpreted as the effect of the gravitational field on the mass–energy of the photon.

Source: Gravitational redshift

Please note light doesn't move slower inside the time dilation bubble. It is one of the tenants of relativity that the velocity of light is the same in all frames of reference. Inside the time dilation bubble, lengths will be contracted too. yes, that does mean things will be smaller on the inside than on the outside. Very, inverted TARDIS of the time dilation bubble.

Let's assume inside the bubble is a table with a ball on it. You push a rod towards the ball to move with it. As the rod enters the bubble, it's time will be dilated and so effectively it slow down. It will also shrink along its length. You cannot move the rod faster than the rod can move inside the bubble. Apparently its mass will have increased, making it harder to move the rod. More effort will be required to move it.

When you withdraw the rod from inside the bubble it will, firstly, increase its length back to its original size and apparently it will be less massive.

The increase in mass inside the bubble isn't strictly speaking directly due to the time dilation. Whatever artificial field is generating the time dilation is causing spacetime to change in a number of ways. In all likelihood, this is due to increasing the local curvature of spacetime itself. This is a dynamic effect, essentially due to spacetime's maintenance of a constant speed of light. Namely, time slows, length contract or shrink, and relativistic mass increases. All these effects go hand in hand. You can't have one without the others.