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Black Holes, Einstein, Time Travel And 6th Formers.

Is it possible to explain the idea of time dilation using A level Physics

Date : 27/09/2012

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Mark

Uploaded by : Mark
Uploaded on : 27/09/2012
Subject : Physics

Black holes, Einstein, time travel and 6th formers.

Is it possible to explain the idea of time dilation using A level Physics?

Most A level Physics courses introduce some of Einstein's equations. For example, the energy equivalence equation: E = mc2 E is energy in joules m is mass in kilograms c is the speed of light in metres per second is introduced as part of nuclear Physics

The photon energy equation: E = hf h is Planck's constant f is the frequency of the radiation is used in the waves and quanta section.

Also the idea of escape velocity is familiar to most A level Physicists: To escape from the surface of the earth into space an object must be travelling at 11 200m/s. To escape from a planet of larger mass or smaller radius would require a greater velocity and in the case of a black hole it is not possible to escape from the event horizon because the velocity required would be greater than the speed of light.

Does time slow down on approaching a black hole? Would we really see a space ship move more and more slowly as it moves towards a black hole and eventually stop entirely at the event horizon? Would the age of the space ship occupants be different from the ages of the people left behind on earth? If the traveller looked back at the earth through a telescope would the people appear to be moving very fast?

The answer to all the above questions is yes, but is it possible to use A level concepts to explain time dilation and why these things happen?

In general, A level students know that the speed of light cannot be exceeded and that the energy of a photon of light is equal to hf, where h is Planks constant and f is the frequency of the light.

When light escapes from the surface of the earth out into space it does work against the earth's gravitational field some energy is converted; there is a red shift, and the frequency of a photon changes.

At the surface of the earth the photon will have energy E = hf

As the photon moves away it must transfer some of its energy to gravitational potential energy to escape from the pull of gravity. Its frequency is therefore reduced, a red shifted photon will have less energy.

? (High gravity) (Low gravity)

Planet, mass M ? = Difference in gravitational potential m = mass of photon

Using conservation of energy

hf1 -hf2 = ?2m - ?1m ( Equation 1 )

?2 and ?1 = gravitational potential energy per unit mass at points one and 2 respectively.

If we make the distance from the earth r = ?, then ?2 = 0

hf1-hf2 = ?m ( Equation 2 )

frequency = 1 ( Equation 3 ) T = time period of the photon. T Putting equation (2) into equation (3) gives

h - h = ?m T1 T2

Then h (T2 - T1) = ?m ( Equation 4) T1 T2

As (T2-T1) approaches zereo

T1*T2 = T1 2 Therefore h (T2 - T1) = ?m ( Equation 5 ) T1 2

Using Einstein's mass energy equivalence equation E = mc2

h = hf1 = mc2 ( Equation 6 ) T1

By substituting into equation 5 we get

mc2(T2 -T1) = ?m which can be rearranged to yield T1

T2 -1 = ? (equation 7)- T1 c2

T2 = 1+ ? (equation 8) T1 c2

T2 = T1 (1+ ? ) got to here c2

T2 is smaller than T1

Showing that time slows down near a large mass.

Having assumed point two is at infinity.

? = Gm (equation 9) r

So T2 = T1 (1+Gm) ( equation 10 ) rc2

Near the sun`s surface: radius = 109 m mass = 2 * 1030 kg got to here

Gm = 6.7 * 1011 * 2 * 10 30 = 1.6 * 10 - 6 rc2 10 9 * 9 * 1016

As the radius approaches 0, then T2 approaches infinity ; and therefore by comparison T1 becomes negligibly small.

We really will observe time stopping near to a point mass. Real black holes have event horizons of finite radius at which time appears to stop. This derivation is only true for weak fields but provides an insight into time dilation effects.

If we think in terms of escaping photons, then the red shift of light attempting to escape from a black hole will be infinite and it will therefore have zero frequency.

T = 1 = 1 = infinity f 0

Time at the event horizon of a black hole is infinite.

There is little point in going to the event horizon of a black hole as it would be impossible to escape. However if we had a very powerful space ship and could remain close but just outside the event horizon then one hour inside the space ship could be equivalent to years or even centuries back on earth.

This could be very useful for someone who is only given a year to live. A trip around a black hole for a couple of hours could mean returning to earth where 100 years have gone by, allowing time for a cure to be discovered!

Compound interest on a few hundred pounds invested for 10 thousand years could also be worth having!!!

However there are several problems with this one way time machine. Firstly the nearest black hole to earth is thought to be in the galactic centre, it would take millions of years for a space ship to reach it. Tidal forces would also rip any ship apart before the time dilation effect was noticed.

It would be interesting to plot a graph of time dilation against distance for a large black hole.

However the equation

T2 = T1(1 + ?) c2

only applies in the `weak-field limit`. That is when the gradient in the gravitational potential d? is small. dr

It cannot be applied near the event horizon of a black hole.

If we apply the equation at the event horizon of a 1 million solar mass black hole then we get.

T2 = T1 (1 + ?) c2

? = Gm G = (universal gravitational constant) 6.7 * 10-11 N m2 kg-2 r M = solar mass 2 * 10 30 Kg, multiplied by 106 = 2 * 1036 kg

To calculate the radius of the black hole we use the escape velocity equation making the escape velocity equal to the speed of light

v2 = 2Gm, r = 2Gm v = velocity of light r v 2 m = mass of the black hole r = radius of the black hole from the centre to the event horizon

r = 2 * 6.7 * 10-11 2 * 1036 (3*108)2

r = 2.9 * 109 metres

Application of T2 = T1(1 + ?) gives T2 = T1(1 + Gm) c2 rc2

Assuming T2 = 1s and making r the event horizon we get

T1 = 1(1+2 *6.7 * 10-11 2 * 1036) 2.9 * 109 (3*108)2

This gives T1 = 1(1+1.02)s

which is clearly incorrect at the event horizon the time should stop.

To calculate what happens near the event horizon requires a more complicated space-time metric and things are different. However the derivation of the weak field equation can give some insight to A level students on why time slows down near large masses.

This time dilation is real. Atomic clocks are the most accurate clocks that can presently be made. Atomic clocks are really light clocks they use the frequency of a very specific wavelength of light emitted from certain atoms to measure time.

The atom emitting the light is affected by the gravitational effect of a black hole.

We are made of atoms so our bodies would also be affected by time dilation.

Mark Williams

Copyright M. C. Williams 2001

Thanks to Dr Jim Al-Khalili department of physics university of surrey for his help with the weak field.

References

A brief History of time Stephen Hawking

In search of the edge of time John Gribben

This resource was uploaded by: Mark