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How to use the Wolfram|Alpha time dilation calculator on Black Holes?

Posted 2 years ago
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Hello,

I’m a graphic artist working for an education company in Arizona. I was given the task of writing a twenty page reader about black holes for middle school students, and all of my research has gone well. But I’ve reached a dead end regarding time dilation. I’m writing a scenario where the reader visits a ten-solar mass black hole while his or her friend stays at a safe distance. I would like to write the following:

If you could stay just in front of the event horizon, you could watch your ten year old friend turn 100 years old in just [xxx *amount of time*].

Unfortunately, I can’t get an adequate answer to this. I was directed to the time dilation calculator here http://www.wolframalpha.com/input/?i=time+dilation+calculator , but I don’t know how to use it. Just playing with it I’ve gotten negative numbers, i, and “exceeds the speed of light’. I have no idea what any of this means. What’s the gravitational acceleration? What’s the rest frame? What’s the radius of what?

I know the time should be very short, but “a blink of an eye” isn’t useful. Would some kind-hearted soul be willing to walk me through this in layman’s language? Or better yet, give me an accurate (but not necessarily precise) number. Any help is greatly appreciated.

Thank you and best regards,

Jack

POSTED BY: Jack M
Answer
9 Replies

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.

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"

    enter image description here

  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:

    enter image description here

Posted 2 years ago

Dear Jofre,

Thank you for your reply. I don't see your response on the website but I did get an email. Unfortunately, I must be doing something wrong. My input is this:

  1. time in rest frame (what or where is the rest frame?): 90 years
  2. gravitational acceleration: 1.52 x 10^12g
  3. radius: 30km

My answer is -1.423 x 10^9 i seconds. I don't know what i means, and I'm not seeing where I wait 63 years to watch my friend age 90 years. It also contradicts an astronomer who only would say I'd wait "less than a blink of an eye." I apologize for the confusion. This subject is a bit thicker than I'm used to :-)

POSTED BY: Jack M
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?

Posted 2 years ago

<<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?>>

First, thanks for your reply and the recommendation of Lawrence Krauss. I will see if I can get in touch with him.

You ask a very good question about how good the book will be. The book (really a reader) is twenty pages and 2000 words, plus pictures. I've been studying black holes for a year now and believe me, there's enough there for 2000 pages even for the average graphic artist. It's very hard to write about this subject without getting into the tall weeds, and some concepts are difficult to wrap the brain around. How do you explain general relativity in two sentences for a 10-year-old? That's the challenge. I'm working on the first draft of the manuscript, which will then go to my editor who's an expert in educational writing. It will then indeed be fact checked and discussed with a physics expert. And then the book will be leveled for the appropriate age, and then, most likely, be fact checked again. It's a long process.

Although I don't have a deep understanding of the physics, I have a reasonable comprehension of the facts and concepts. I do understand time dilation, the equivalence principle, and even a bit of the Lorentz transformation, but for the finer mathematical details I get lost in the equations. And yes, I did watch all two hours of Dr. Physics. Even grasped some of it :-) https://www.youtube.com/watch?v=foRPKAKZWx8

The purpose of the books isn't to provide a comprehensive study but an introduction. Hopefully, kids (and maybe even adults) will find black holes and relativity as fascinating as you and I do, and inspire "Hey Dad, I bet you didn't know about this . . . !!!"

Thanks again for your help and best regards,

Jack

POSTED BY: Jack M
Answer

I would recommend the book: Edwin R. Taylor and John Archibald Wheeler, Exploring Black Holes: Introduction to General Relativity, 2000, Addison Weslay Longman. Specifically, pages 2-28 to 2-31 discuss gravitational effects on time.

In the following dt[Infinity] is a time interval measured by a stationary clock at infinity. dt[r,M] is the corresponding time interval measured by a clock at a relative radius r to the radius of the black hole horizon. So putting r==2M would correspond to being at the black hole horizon.

dt[r_, M_] := dt[Infinity] Sqrt[1 - 2 M/r]

The following plots the time interval of a clock at radius r relative to the interval measured by a clock at infinity. enter image description here

The time interval goes to zero at the black hole horizon (r = 2) so you can get values as small as you want. For example, at r = 2.0001 we obtain:

dt[2.0001, 1]
0.00707089 dt[\[Infinity]]

As the clock is moved further and further out its time intervals get closer and closer to the clock at infinity.

Posted 2 years ago

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 ;-)

POSTED BY: Jack M
Answer

Yes, that's how many seconds would pass on the clock that near the event horizon compared to 1 second passing on the far clock. And even less time would pass as you got closer. At r = 2 no time would pass. But to make a good book you should get the help of an interested physicist. Maybe like Lawrence Krauss at Arizona University.

And I bought that book for $37.50 at Border's Book Store sometime after 2000.

Posted 7 months ago

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?

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.

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