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I am in the midst of writing a book wherein one of my characters (Let's call her X) can affect spacetime, mostly by causing time to dilate.
My character is earthbound and a shadow organization discovers that she has these powers because she dilates time to recover from a personal tragedy. The issue becomes, how does the organization realize that she can dilate time in the first place? Would the characters on earth realize that time has been dilated or they will be oblivious to the same?
If someone can make time dilate, this must done in a way that this ability useful. It shall be assumed in this answer this time dilation power only works on people and things in the immediate vicinity of the character with the power. If her power encompassed the entire planet earth nobody would notice anything had changed with the possible exception of astronauts returning from deep space.
But let's start with the not so useful version of her time dilation power.
If, in contrariwise case, she made time in her vicinity and if this includes herself, then while she and people nearby wouldn't be aware their time was dilated, i.e., moving more slowly than for people and things outside her field of influence.
The shadow organization might her doing this by observing one of their nefarious operations going wrong by her accidentally dilating time. If this happened several times and she was always present this fact would enable them to discover her and her powers.
She could employed to slow down time in specific areas while the shadow organization does its nefarious deeds.
But that was the not so useful case. Now for the more useful case. This time her time dilation power slows down time people and things in her vicinity but she remains unaffected and moving her normal rate of speed. Any shadow organization that accidentally crossed her path while she was in time dilation mode would observe everything around he was time dilated but she was still moving normally. Your average criminal mastermind should have no difficulty deducing what was going on, and put her on their payroll or coerce her into using her power for evilness.
This answer has sketched two possible modes by which a time dilation power might work and its affects. Like ChrisM, she could be detected by the affects of her time dilation. The likelihood of her detection will be increased if the character and the shadow organization cross each other's paths.
Disclaimer: it's been awhile since I've had any instruction in relativistic physics.
That being said, time dilation is something observed based on general relativity - a clock traveling at relativistic speeds (any appreciable fraction of the speed of light) relative to a stationary observer will appear to take longer than 1 second to tick. Time dilation is also mutual - two observers moving at relativistic speeds relative to one another will observe time passing for the other at a slower rate.
Regarding your character and your comments on Adi's post, it would be observable to a stationary person (or object) that your character was experiencing time dilation (whether via geosynchronous satellite feed that can see her, or CCTV, or some such recording device). She would appear to be motionless and ageless as the world progressed around her - perhaps for many years or decades.
Edit: to answer your original question, if X were to dilate time using a means known to cause relativistic effects (for example, moving really fast), she would see time moving very slowly around her. But observers moving in "normal" time would see her as progressing very slowly through time.
Edit in response to comments: Special relativity causes time dilation in real-world physics, and relies on the fixed speed of light in space and rapidly-changing distances. However, assuming your character is able to create the same time-dilating effects as special relativitiy without the usual cause, it's not unreasonable that it would still follow the same principles of mutual observance.
Of course, your character is defying physics so pretty much anything goes and you can handwave the effects to suit your needs.
For the system or person affected by the time dilation, it will not realize time has been dilated.
For the system or person outside of the affected area, it will observe time has run slower within the area.
It is similar to Twin Paradox, where the affected person is the twin that do the space travel (near speed of light), and affected by the time dilation, whereas the outside observer is the brother who has aged much more.
For the sake of imprisonment, depending on how you want the punishment to be:
If she distorts time in the way that gravity does according to general relativity, she could make time pass faster for her than others, which sounds like what you're looking for. From the perspective of an outside observer, she would appear to be moving very fast. Alternatively, if she were sitting still and wearing a watch, the person outside might not notice a difference in her movement speed, but might realize that her watch was running several hours ahead because it was inside the distortion field with her and experienced a bunch of "extra hours" as compared to a clock outside the field.
By contrast, the time dilation associated with special relativity involves people moving quickly relative to each other without acceleration or gravity. In case, two people approaching each other at nearly the speed of light carrying clocks would both observe the other person's clock to be running slower than their own, and there would be no way for them to tell the difference.
You need to draw space-time diagrams (see this post for references) with the world lines of the parties involved. From there you can figure out what happens in various cases, and concoct interesting scenareos.
But if time is dialated without going anywhere, it might be simpler to understand it as a time slowing field that affects that object. Draw tick marks on the world lines to show each one’s proper time including your time-slowing power. Again, you can read off the interactions from such diagrams, to find out what each party experiences.
Your idea is the opposite sign of this one so some of the ideas and background in the Answers may be useful to you.
