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.

Extremely massive astronomical object. Extremely strong gravitational pull Nothing can escape from below the event horizon.

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

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.

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.

GR equations in 2-D give 4 coupled differential equations. Solve using Runge- Kutta 4 th, 5 th order method (ode45 in MATLAB). Increasing complexity

General relativity reduces to the classical picture in flat space time. Solving equations in the case of circular motion.

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

Not surprisingly, our observer falls into the black hole. Not so good for our observer.

With constant acceleration, 3 things can happen: 1. Acceleration is too low: Not so good for our observer.

2. Acceleration is too high: Not so good for science.

Or if the acceleration is just right….

The acceleration will just cancel out the gravitational pull.

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.

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*10 9 greater than the sun. We start from 150 times away from the event horizon.

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.

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This journey is more physically realistic. We can also model the acceleration experienced during this journey. This poses some problems.

Acceleration Component at (blue) Acceleration Component ar (red) Total Acceleration Acceleration measured in g 150 3

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/s 2.

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.

Try to have change happen gradually. In particular, a smooth transition from falling inwards to beginning to escape.

Though the path may look smooth, the rapid changes in g show it is not really. 3 3

Depends on who we want to send. ScientistsFighter Pilots 1-2g 6-9g

An improvement, but it still kills them. 3 3

But obviously we don’t get as close. 3 3

Starting at 4.4*10 13 m corresponds to experiencing around 1g when hovering. It is impossible to go much lower because hovering acceleration alone would be too many g.

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.

We’d like to extend thanks to: –Our supervisor, Geraint Lewis. –TSP co-ordinator, Dick Hunstead.

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 Serway, Moses, Moyer, Modern Physics, (California, USA, 2005 (3 rd Edition), Thomson, Brooks/Cole). Wikipedia, G-forces, Black Holes, (2009, Wikipedia). All graphs produced using MATLAB7. (2009, Mathworks Inc.). Images from: – – –