Black Holes

APPM 2360 Fall 2012
Lab/Project 1

 

"... in all my life I have not labored nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now."

- Albert Einstein

Black holes are some of the most mysterious objects in the universe. Theories suggest that these cosmic vacuum cleaners may provide passages to other universe or act as time machines. Even though the mathematical theories behind black holes are complicated, one can analyze one of the properties of a black hole with only a few weeks of ODE experience.

 

1. Instructions

This lab is due Monday October 8, 2012 at the beginning of lecture. TA's will hold their regular office hours in ECCR 143(computer lab) during that week before it is due. The TAs in the lab are there to help you with Mathematica and Matlab syntax; they are not there to debug your files.  Please keep this in mind.

  On your title page, clearly mark

Format is worth 10% of your grade. Please refer to the following writing guidelines for the expository sections of this report. You are required to know and follow the Writing Guidelines for all labs. Guidelines and good and bad sample labs can be found on the web page for APPM 2460.

DO NOT LEAVE ANY OF YOUR FILES SAVED ON THE COMPUTERS IN THE LAB!
IF YOUR FILES ARE FOUND SAVED ON A LAB MACHINE,
IT MAY RESULT IN YOUR GROUP GETTING A 0 (ZERO) FOR THE LAB.

You can either E-mail them to yourself, use webfiles, or save on a USB 'memory stick'

For this lab, you will be using Matlab.  Matlab is installed on the computers in computer labs in ECCR.


2. Background

Stars are born, age and die just like humans. A black hole is an astronomical object predicted by Einstein's relativity theory and is created when certain stars run out of fuel and “die”. Black holes are regions in the universe with extremely high density and a very strong gravitational pull on the surroundings. Not even light has the power to escape the surface of a black hole. The density of a black hole can be compared to a grain of salt having the mass of approximately 20,000 cars.

Due to the enormous gravitational pull, black holes suck nearby material, e.g., nearby stars, into the hole. The effect of a black hole is so strong that it even affects the time around the hole. This leads to some interesting effects, one of which you will investigate in this lab.

Imagine that a space probe is launched towards a black hole from a space station. According to Einstein's relativity theory, a passenger inside the probe will see that he or she is sucked into the black hole. However, the theory also predicts that an observer on the space station will never see the probe disappearing into the hole!

To an observer on the space station, it will seem like the probe approaches the surface of the black hole slower and slower without ever reaching the surface. This effect is caused by the black hole distorting the time of its surroundings.

This is indeed mind-boggling and one of the strangest phenomena predicted by physics!

3. Problem Statement

Now let us focus on your mission. Imagine that one day an astronomer comes up to you with a differential equation that she wants you to solve in order to verify a hypothesis she has based on his knowledge of black holes.

*** Hypothesis:
Imagine a space station at a fixed distance of 4 Galagamile radius units(GRUs) away from the center of a black hole (see Figure 1 below). The black hole has radius 2 GRU. The space station launches a probe towards the black hole. An observer on the space station follows the probe with a telescope as the probe approaches the surface of the black hole. The hypothesis is that according to the observer on the space station the probe never reaches the surface of the black hole.

Figure 1:

The astronomer has formulated an IVP that models the position of the probe as observed from the space station as a function of time.

Your task is to verify the hypothesis by analyzing the following initial value problem.

Equation (1):

The independent variable is time (t) measured in seconds from when the probe is launched from the space station. The dependent variable is distance (x) measured from the center of the black hole. Note that we impose for this problem that x > 2.

In order to analyze this problem, you need to discuss the following issues in your report:

  1. Classify the differential equation (1) in terms of linearity, and order, and if linear determine homogeneity and if this is an autonomous or non-autonomous equation.
  2. What is the initial velocity of the probe (Remember units, ie GRU/sec)? Without solving the initial value problem (IVP) in equation (1), determine if the distance between the probe and the black hole is increasing or decreasing. (Justify your answer)
  3. Does equation (1) have a unique solution through the given initial condition? (Justify your answer)
  4. Find the equilibrium point(s) of (1), and classify each as stable, semi-stable, or unstable. Describe what this equilibrium signifies in a physical sense.
  5. Determine the long term behavior of the solution to (1). Use a vector field plotter (if you need help see worksheet 3 on the 2460 website, or use Dfield below ) and your answer to question 4. 
  6. Does the probe ever reach x = 2? Why or why not?
  7. Solve the initial value problem (1) analytically using separation of variables. You do not have to solve explicitly for x.
  8. Using MATLAB and the BlackHoleEuler program, find a numerical solution to equation (1) in the time interval 0 <  t  < 30. Use several different time steps: h = 0.01, 0.1, 0.5, 1, 2, 5 and 10. Include plots of the results (but not endless streams of numbers) in your write up!
  9. Describe what happens as the time steps change. Are any of the time steps you used too large? Explain.
  10. Assume the true value of x(30) is 2.013297589350489. Determine the error in approximating the value of x(30) with a stepsize of h=0.01. Use the MATLAB code BlackHoleRK4 to determine the largest value of h taht will provide the same accuracy as Euler's method with h=0.01. Explain why these values are so different.
  11. Plot the numerical solution of x'(t) =-(1/2-1/x)2x-1 with initial conditions x(0) =2 and x (0) =4 on the same graph using a time step of h = 1. Look carefully at the plots and the numerical values generated. Do the solutions on this plot violate uniqueness? (HINT: If solutions of the same differential equation starting from different initial conditions ever intersect, then uniqueness is violated.)
  12. Use Picard's Theorem to support your answer for question 11 analytically.
  13. Compare your conclusions regarding long term behavior from your analysis in questions 1-6 with your numerical results in questions 8-9.  Do they agree? Why or why not?
  14. How does the stepsize affect the computed value x(30)? Which is more accurate?
  15. Does the observer on the space station ever see the probe reach the black hole?

4. Comments

  1. Do include a printed version of your code in the printed copy of your write up that you hand in.  DO NOT include pages of number-output from these programs.  You can (and should) include PLOTS of output from these programs.
  2. Be sure to discuss the hypothesis (***) given in the lab, and prove or disprove it in your write-up.

5. Interesting links

  1. Pplane and Dfield. This is a link where you can find some interesting mathematical tools that may be useful for this lab.
  2. For some more information about black holes, click here.

Created: June 1999 by Kristian Sandberg
Last updated: September 12, 2012 by Josh Snyder