Wrist watch close to a black hole The article in the source below says,

"As you get closer to a black hole, the flow of time slows down..."

I am trying to understand this whole time slowing down with the following thought experiment/question:
How would a regular wrist watch respond to getting closer to a black hole? Does it literally tick slower? Do the digits on a digital wrist watch that is closer to a black hole have its digits change very slowly?
( I come from an Electrical Engineering background. Would very much appreciate it, if at all possible, if you can please answer me using my background as a reference )
Sources:
https://lweb.cfa.harvard.edu/seuforum/bh_whatare.htm
 A: It depends on what you mean when you say "does it literally tick slower". More specifically, does it tick slower in comparison with what?
The Thought Experiment: what happens when you fall down a black hole?
Let us first assume that you are falling alone on the black hole for whatever reason. Do you notice anything weird with your watch? No, you do not. As mentioned in other answers, time has literally slowed down, and that applies to yourself, not only to your watch.
On the other hand, suppose you jumped off a spaceship and the crew over at the spaceship are monitoring you. In order to do this, they equipped you with a device that sends a radio signal whenever your wristwatch ticks. Hence, you are sending them a signal per second. What do they notice as you get closer to the black hole?
At first, they will not notice anything strange. They'll keep receiving signals at intervals of roughly one second, for the effects are quite small while you both are somewhat far from the black hole and close to each other. As you go near the black hole, they'll notice your signals are taking longer to come in. Instead of receiving one signal per second, they are receiving one signal every two seconds, eventually one signal every three seconds, and so on. In other words, your watch is ticking slower with respect to the crew over at the spaceship.
This effects gets stronger and stronger as you get close to the black hole. If someone from the crew takes a look at you through the window, they'll notice you are also getting redder, due to an effect known as gravitational redshift (in short, the light that's being reflected by your body loses energy as it moves away from the black hole, and light losing energy means light color getting shifted to the red). Eventually, this redshift effect gets so strong they can no longer see you (light has been shifted to the infrared, which our eyes can't see) and when they took a look at your signal (the frequency of which has also been changing due to the same reason) they'll notice the intervals have become so large they are no longer receiving your "messages" anymore. In spite of that, they've never seen you falling into the black hole, only getting closer and closer.
What about you? Did you get into the black hole? Yes. Within finitely many ticks of your clock, you will get into the black hole (and there will be nothing particularly shocking once you get into it, there is no sign saying "Welcome to the black hole!" nor any experiment you can perform locally to decide where is the black hole boundary). Hence, for you, time hasn't slowed down, and everything seems just as normal as ever. Nevertheless, if you pick a telescope and peek at a clock on the spaceship you jumped from, you will notice their clock seems to be running ridiculously fast.
It is in this sense that time slows down in a black hole. The particular mechanism of the clock does not matter: it is time itself that is slowing down, not just the clock. Hence, no one can perceive the effect locally, one has to compare their own wristwatch with a clock somewhere else to notice time is slowing down for them.
How do clocks know they should slow down close to a black hole?
This is in fact a consequence of the fact that gravity is just a manifestation of spacetime curvature. Space and time are actually a single entity which can bend and stretch and so on, and the consequences of this bending and stretching are what we call gravity. The clocks don't know they should tick slower in the same way you don't voluntarily move slower when you are in strong gravitational fields. As far as you, or the clock, are concerned, you are behaving as usual. The tricky thing is that "behaving as usual" has different meanings depending on where you are in spacetime.
I'll add another paragraph reinforcing this point because it is really important: it is not the watch that slows down, it is time itself. It doesn't matter what the watch is made of, it doesn't matter whether it is digital or analogue, or if it consists of you counting periods of radiation of a Caesium-133 atom, or if it consists of
counting how many times a beam of light has reflected on a pair of mirrors, or if you are measuring time with your heartbeat, or if you are clapping in regular intervals, or if you are repeatedly listening to your favorite song and counting how many times it has played. In every situation the effects are going to be the exact same because it is not the watch or the mechanism that is being slowed down with respect to the clock at the spaceship, it is time itself.
The fact that time runs differently in different regions of spacetime is, quite literally, the cause of gravity. This is beautifully discussed and illustrated on the PBS Spacetime video Does Time Cause Gravity?.
A: The answer is Yes, a wrist watch, regardless of if it is mechanical or electrical, runs slower near a black hole.  It also runs slower near the Earth, or the Sun, or any planet or even dark matter than it does in "open space".
All of these things slow down time.  But it is vital for you to understand that if you are wearing the wrist watch, then you would not notice the slowing of time because you are also subject to that same slowing of time. It is ONLY if you are looking at the wrist watch from a very far distance that you will notice that it is running slower than the time at your location.  And in answer to your other question, everything including every electrical circuit that you can think of also runs at that slower time.
This has two major implications,  first; many scientists believe that this time dilation is responsible for gravity. This is the warping of spacetime that you have heard about. That is, things fall to where time runs slowest. So things fall to the Earth, or the moon, or the sun or a black hole because time runs slower on the surface of these than it does in space.
Second, if you want to travel to the future, the process is very simple.  If you can get yourself to a black hole and orbit it for a few years in your own time, you can come back to Earth in the distant future to visit your great great great grandchildren.
A: Let's say a capacitor has energy 10 J. To make that energy do work for us we may tie the capacitor on a string and then lower it towards a black hole. The electric energy does work when it pulls on the string. Work done when the capacitor is at the lowest possible position is 10 J.
A voltmeter with long wires will show that voltage of the capacitor is decreasing when it is being lowered. From that we may conclude that the capacitance is increasing.
Same story with an inductor:
Let's say an inductor has energy 10 J. To make that energy to do work for us we may tie the inductor on a string and then lower it towards a black hole. The electric energy does work when it pulls on the string. Work done when the inductor is at the lowest possible position is 10 J.
An ammeter with long wires will show that current of the inductor is decreasing when it is being lowered. From that we may conclude that the inductance is increasing.
Now finally as we have established that inductance and capacitance are inversely proportional to gravitational potential we can calculate the frequency of a LC-circuit near a black hole: $1/\sqrt{L_{down}C_{down}}$
The zero point of gravitational potential must be set at the event horizon for this to work.
A: The watch becomes assimilated by the black hole. The watch’s process is overcome by the black hole's process. The watch is forced to participate in the black hole’s process. At the event horizon, the assimilation is complete. The watch’s tick rate is related to the gravitational time dilation.
You could say that the probability of a process occurring is equal to $\sqrt{1 - \frac{2GM}{rc^2}}$
