Hello,

I am also trying to use the time dilation tool for a supermassive black hole, specifically, TON 618, which has approximately 66 Billion solar masses. However. I continue to get imaginary numbers. Here is the rundown:

So this gives us a horizon radius of 1.949x10^14 m, or 1.949x10^11 km

Converting to g, this gives us 23.515 g (doesn't this seem really low?)

Entering these figures, I keep getting imaginary numbers. Does I make a mistake somewhere along the line? Thanks in advance.

Dear David,

Thank you for your reply. Unfortunately, I need further clarification here too. What does 0.00707089 correspond to? Is that how many seconds of my time near the black hole passes relative to my friend's at a safe distance?

I looked into your book on Amazon ($175!). Remember -- I'm a graphic artist and haven't done algebra in twenty years. I can tell you the difference between Helvetica and Helvetica Neue, or make you a darn good logo ;-)

Hi Jack, the answer to you question is 63.6 years. You can get easily this value using Wolfram|Alpha:

1. Ask Wolfram|Alpha for "black hole 10 solar masses"

   

2. Then go to the Wolfram|Alpha gravitational time calculator and insert the previous values for the horizon radius, the surface gravity and 90 years (time that your friend turns 100) and you will get the answer:

   

You want to know the effect that a black hole has on time. The idea of rest frame is not at issue here. All relevant clocks are at rest. Imagine an infinite pole extending radially from the event horizon of a black hole. At places along the pole you mount clocks and also mark the radial distance of the clock from the center. The radial distance will be in terms of the radius of the event horizon of the black hole.

Identical clocks, wristwatches say, will run at different rates depending on their relative motion (not relevant here) and on what gravitation field they are in. The clocks at different radii on the pole will run at different rates and you can calculate that from the formula I gave you in my other answer. If we wanted to pick a standard clock we might pick the one at infinity. Why? Because it is in a region of space where the gravitational field is zero, which means flat spacetime. That might be the equivalent of 'being at rest' for the black hole problem. The formula I gave in my other answer gives the interval of time that will pass on one of the clocks on the pole as a fraction of the time that would pass on the 'clock at infinity'. If you look at the clock at the event horizon, no time will pass. Or you can make it as short as you want by getting close enough to the radius of the event horizon.

This whole discussion is getting a bit off the topic of Wolfram Language or Wolfram Alpha. If you know very little about general relativity (even though you might be a great graphic artist!) how can you write a small book about black holes for middle school students? Are you partnering with a physicist who is well versed in gravitational physics? Are you an expert in that yourself? If not, how good the students get a good book?

A good starting point is On Continued Gravitational Contraction by J. R. Oppenheimer and H. Snyder. If I remember it correctly that is the nice paper where light cones fall graceful along the geodetic lines of the metric of the black hole into it, explaining the event horizon virtually by means of illustration ... a point that could be appealing to a graphic artist.