Falling into a black hole by Alison Hammond and Jason Cheng
Background-Gravity • Gravity is one of the four fundamental interactions. • General Relativity (GR) is the modern interpretation of gravity. • GR says gravity is not a force! It is due to following paths of least resistance on a curved space-time.
What is a black hole? • Extremely massive astronomical object. • Extremely strong gravitational pull • Nothing can escape from below the event horizon.
How do we study black holes? There are three ways we could study black holes. 1. We go to a black hole • Too far away, the closest black hole is some 1,600 light years away • Too dangerous, if anything goes wrong, one goes in, and will never come back out 2. We create a black hole • CERN is working on it, let’s do something else before they actually make one
How do we study black holes? 3. We simulate a black hole using GR equations • It would take more than 1000 years to do this numerically by hand. • So we use our good friend MATLAB • Modeling in 1 space and 1 time dimension • Use MATLAB to solve the differential equations.
Aims of the Project • Using MATLAB: -Explore motion near a black hole. -Investigate objects falling into a black hole. -Analyse the effect of radial acceleration. -Model explorers approaching the black hole and then coming back out.
Method • GR equations in 2-D give 4 coupled differential equations. • Solve using Runge-Kutta 4th, 5th order method (ode45 in MATLAB). • Increasing complexity
Circular Motion • General relativity reduces to the classical picture in flat space time. • Solving equations in the case of circular motion.
Schwarzschild metric • Equations of GR near a black hole • Describes the space-time geometry near a non-rotating, non-charged black hole (Schwarzschild black hole) • In our project, we ignore angular terms
No Acceleration • Not surprisingly, our observer falls into the black hole. • Not so good for our observer.
Constant Acceleration • With constant acceleration, 3 things can happen: 1. Acceleration is too low: • Not so good for our observer.
Constant Acceleration 2. Acceleration is too high: • Not so good for science.
Constant Acceleration Or if the acceleration is just right….
Hovering • The acceleration will just cancel out the gravitational pull.
Changing Acceleration • Now let’s look at when acceleration changes. • Introduce ‘k’ into equations. The acceleration can now take, for instance, a functional form over time. • N.B. ‘k’=1 is hovering acceleration • We want to get close to the black hole and investigate.
Physical Interpretation • Note about scaling factors – MATLAB solves the equations for the case m=1. Scaling factors were then calculated and used to give correct units for realistic masses. • This diagram models a super-massive black hole –it has a mass 1*109 greater than the sun. We start from 150 times away from the event horizon.
Controlled Fall and Escape
Physical Interpretation • So we have an approximately 2-month mission in the vicinity of a black hole. • The very sudden change brought about by our function, however, is quite physically unrealistic. • A smooth functional form gives a better picture.
Controlled Fall and Escape 3
Physical Interpretation • This journey is more physically realistic. • We can also model the acceleration experienced during this journey. • This poses some problems.
Controlled Fall and Escape 150 Total Acceleration Acceleration Component at (blue) Acceleration Component ar (red) Acceleration measured in g 3
What are g-forces? • G-force is the acceleration experienced by an object relative to free-fall. • G-forces are measured in multiples of the acceleration we experience at the earth’s surface: 1g=9.8m/s2.
The problem of Survival • Humans cannot survive high g-force levels. • Our current model, with g-forces of 150g, is clearly going to kill whatever observers we send. • This is a problem.
How do you minimise g-forces? • Try to have change happen gradually. • In particular, a smooth transition from falling inwards to beginning to escape.
How do you minimise g-forces? • Though the path may look smooth, the rapid changes in g show it is not really. 3 3
What g-forces do we need? • Depends on who we want to send. Scientists Fighter Pilots 6-9g 1-2g
Results • An improvement, but it still kills them. 3 3
Results • But obviously we don’t get as close. 3 3
Discussion • Starting at 4.4*1013 m corresponds to experiencing around 1g when hovering. • It is impossible to go much • lower because hovering • acceleration alone would • be too many g.
Further Investigation • The next logical step would be to investigate the same problem in two spatial dimensions with time. • Another possibility is to investigate smaller black holes. • The procedure is identical, but the maths much messier and more time consuming.
Acknowledgments • We’d like to extend thanks to: • Our supervisor, Geraint Lewis. • TSP co-ordinator, Dick Hunstead.
References • Griffiths, David, Chapter 12: Electrodynamics and Relativity, Introduction to Electrodynamics, (San Francisco, USA, 2008: Pearson, Benjamin Cummings). • Hartle, James B., Gravity: An Introduction to Einstein's General Relativity (USA, 2003: Pearson, Addison Wesley). • Lewis, Geraint & Kwan, Juliana, ‘No Way Back: Maximising Survival Time Below the Schwarzschild Event Horizon’, Publications of the Astronomical Society of Australia, 2007, 24, p. 46-52. • Serway, Moses, Moyer, Modern Physics, (California, USA, 2005 (3rd Edition), Thomson, Brooks/Cole). • Wikipedia, G-forces, Black Holes, (2009, Wikipedia). • All graphs produced using MATLAB7. (2009, Mathworks Inc.). • Images from: • http://jcconwell.files.wordpress.com/2009/07/black_hole_milkyway.jpg • http://app.ucdavis.edu/algebra/blackhole3.jpg • http://lgo.mit.edu/blog/drewhill/files/blackhole.gif