special relativity david berman queen mary college university of london

Special RelativitySpecial RelativityDavid BermanDavid Berman

Queen Mary CollegeQueen Mary College

University of LondonUniversity of London

Symmetries in physicsSymmetries in physics

The key to understanding the laws of nature is to The key to understanding the laws of nature is to determine what things can depend on. determine what things can depend on.

For example, the force of attraction between For example, the force of attraction between opposite charges will depend on how far apart opposite charges will depend on how far apart they are. Yet to describe how far apart they are I they are. Yet to describe how far apart they are I need to use some coordinates to describe where need to use some coordinates to describe where the charges are. The coordinate system I use the charges are. The coordinate system I use CANNOT matter. CANNOT matter.


Let’s use Cartesian coordinates.Let’s use Cartesian coordinates.

Object 1 is distance x1 along the x-axis and Object 1 is distance x1 along the x-axis and distance y1 along the y-axisdistance y1 along the y-axis

Object 2 is distance x2 along the x-axis and Object 2 is distance x2 along the x-axis and distance y2 along the y-axisdistance y2 along the y-axis

The distance squared between the two objects will The distance squared between the two objects will be given by:be given by:

222 )21()21( yyxxd


The force is inversely proportional to the distance The force is inversely proportional to the distance squared.squared.

What transformations can we do that will leave the What transformations can we do that will leave the distance unchanged?distance unchanged?

Translation: Translation:

byy

axx

byy

axx

22

22

11

11


Rotations:Rotations:

2)cos(2)sin(2

2)sin(2)cos(2

1)cos(1)sin(1

1)sin(1)cos(1

yxy

yxx

yxy

yxx


We can carry out the transformations We can carry out the transformations described of translations and rotations and described of translations and rotations and yet the physical quantity which is the yet the physical quantity which is the distance between the two charges remains distance between the two charges remains the samethe same..

That is a symmetry. We carry out a That is a symmetry. We carry out a transformation and yet the object upon transformation and yet the object upon which the transformation takes place which the transformation takes place remains the same or is left remains the same or is left invariantinvariant. .


The important quantities in physics are those The important quantities in physics are those that are that are invariantsinvariants. That is the things that don’t . That is the things that don’t transform.transform.

Other things will change under transformations Other things will change under transformations and so will depend typically on our choice of and so will depend typically on our choice of description, for example which way we are description, for example which way we are facing.facing.

Space and TimeSpace and Time

We live in both space and time. There are the We live in both space and time. There are the usual three dimensions of space we are used to usual three dimensions of space we are used to and also one more of time. We perceive time as and also one more of time. We perceive time as being very different to space though.being very different to space though.

How different is it really? How different is it really?

To arrange a meeting I need to specify a time To arrange a meeting I need to specify a time and a place. I can describe the place by using and a place. I can describe the place by using some coordinates and the time by specifying the some coordinates and the time by specifying the hour of the day (that’s just a time coordinate).hour of the day (that’s just a time coordinate).


Distances in space can given as we have shown.Distances in space can given as we have shown.

Distances in time would also be given by the difference Distances in time would also be given by the difference of the two times that is:of the two times that is:

12 ttt

12

12

yyy

xxx


How do we add up distances in different How do we add up distances in different directions?directions?

We’ve already seen that it is NOT just the sum We’ve already seen that it is NOT just the sum of the distances in the different directions rather of the distances in the different directions rather the total distance is given by:the total distance is given by:

222 yxd


How do we find the distance in space-time. That is How do we find the distance in space-time. That is given the distance in space and the distance in time given the distance in space and the distance in time how can we combine them to give the total distance how can we combine them to give the total distance in space-time?in space-time?

Wrong guess:Wrong guess:

222 dts


Einstein had a better Einstein had a better idea.idea.

He combined space He combined space and time found the and time found the right way to describe right way to describe distances in distances in spacetime.spacetime.

EinsteinEinstein

In 1905, while working as a patent office clerk in In 1905, while working as a patent office clerk in Bern, published his work on special relativity. His Bern, published his work on special relativity. His insights in that paper were essentially that space insights in that paper were essentially that space and time should be combined in one thing, and time should be combined in one thing, spacetime. He also realised the right way to spacetime. He also realised the right way to construct invariant distances in spacetime.construct invariant distances in spacetime.

The same year he also published two other key The same year he also published two other key papers in other areas of physics. It really was an papers in other areas of physics. It really was an enormous break through year for Einstein.enormous break through year for Einstein.

SpacetimeSpacetime The distance in spacetime is given by:The distance in spacetime is given by:

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SpacetimeSpacetime

When we measure distances we use the same When we measure distances we use the same units for x and y. If we didn’t then we could units for x and y. If we didn’t then we could convert between units in the distance formula convert between units in the distance formula like so:like so:

With w the ratio of the two different units. With w the ratio of the two different units. Instead we pick w=1 and use the same units for Instead we pick w=1 and use the same units for

our x and y distances.our x and y distances.

2222 ywxd

SpacetimeSpacetime

For spacetime, what is the choice of units of time For spacetime, what is the choice of units of time that will set w=1 and give us the equivalent unit that will set w=1 and give us the equivalent unit for time as for space?for time as for space?

If we measure space in meters then we should If we measure space in meters then we should measure time in light meters. (More about this measure time in light meters. (More about this later).later).

SpacetimeSpacetime

Given that the distance in spacetime is given Given that the distance in spacetime is given by:by:

What are the transformations that leave this What are the transformations that leave this distance invariant? What is the symmetry? That distance invariant? What is the symmetry? That is how can we transform space and time so that is how can we transform space and time so that the distance in spacetime remains the same.the distance in spacetime remains the same.

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SpacetimeSpacetime

LorentzLorentz realised that realised that there was a symmetry there was a symmetry in nature where you in nature where you could transform space could transform space and time distances in and time distances in the following way.the following way.

Lorentz TransformationsLorentz Transformations

21

1

)(

)(

vb

vxtbt

vtxbx


Spatial distances can shortenSpatial distances can shorten Time distances can also shortenTime distances can also shorten The spacetime distance is the same that is it is The spacetime distance is the same that is it is

invariant under these transformations.invariant under these transformations. v is a velocityv is a velocity Units are chosen such that time is measured in Units are chosen such that time is measured in

light meters.light meters.


Distances in space will depend on the velocity of Distances in space will depend on the velocity of the observer.the observer.

Distances in time will depend on the velocity of Distances in time will depend on the velocity of the observer.the observer.

This is just like saying that spatial distance in This is just like saying that spatial distance in one direction depends on which way you are one direction depends on which way you are facing.facing.

The equivalent to the angle you are facing is The equivalent to the angle you are facing is velocity you are moving at. velocity you are moving at.

ExperimentsExperiments

Thousands of Thousands of experiments have experiments have been done checking been done checking the Lorentz the Lorentz transformations and transformations and the altering of time the altering of time and space depending and space depending on velocity.on velocity.


Lifetime of elementary Lifetime of elementary particlesparticles

Orbiting atomic clocksOrbiting atomic clocks Collider physicsCollider physics Michaelson Morley Michaelson Morley

experiment: Speed of experiment: Speed of light is constant no light is constant no matter what your matter what your velocityvelocity


In the experiment carried out by Michaelson and In the experiment carried out by Michaelson and Morley an attempt was made to measure speed Morley an attempt was made to measure speed of light parallel to the motion of the earth and at of light parallel to the motion of the earth and at right angles to the motion of the earth.right angles to the motion of the earth.

According to our usual notions of how velocities According to our usual notions of how velocities add there should have been a difference.add there should have been a difference.

They found the speed of light was the same They found the speed of light was the same whether it was directed alongs the earth’s whether it was directed alongs the earth’s motion or not. This agrees with relativity, the motion or not. This agrees with relativity, the speed of light is the same no matter how fast speed of light is the same no matter how fast you are going!you are going!

ConsequencesConsequences

How big is a light meter?How big is a light meter? Speed of light is about 300000000m/sSpeed of light is about 300000000m/s One light meter is about 0.0000000033333 sOne light meter is about 0.0000000033333 s To convert to velocities measured in m/s we To convert to velocities measured in m/s we

need to divide by c- the speed of light a big need to divide by c- the speed of light a big number.number.

Most velocities in every day are much much less Most velocities in every day are much much less than the speed of light which is why we don’t than the speed of light which is why we don’t notice the Lorentz transformations in ordinary notice the Lorentz transformations in ordinary life.life.


Notice that v/c can’t be 1 or the Lorentz Notice that v/c can’t be 1 or the Lorentz transformation become infinite and time transformation become infinite and time and space become infinitely transformed.and space become infinitely transformed.

We can’t travel faster than the speed of We can’t travel faster than the speed of light.light.


Just as space and time Just as space and time rotate rotate into each other so do into each other so do other physical quantities. What matter is the invariant other physical quantities. What matter is the invariant quantity.quantity.

Energy and Momentum also transform into each other Energy and Momentum also transform into each other under Lorentz transformations. The invariant quantity is:under Lorentz transformations. The invariant quantity is:

222 pEm


Putting back in c, the speed of light so that energy and Putting back in c, the speed of light so that energy and momentum would be measured in SI units this equation momentum would be measured in SI units this equation becomes:becomes:

If p is zero we get the celebrated equation:If p is zero we get the celebrated equation:

22242 cpEcm

2mcE

ConclusionsConclusions

Space and time should be combined to Space and time should be combined to spacetime a single entity.spacetime a single entity.

The invariant measure of distance on spacetime The invariant measure of distance on spacetime isis

With the unit the light meter.With the unit the light meter. Lorentz transformations leave this distance Lorentz transformations leave this distance

invariant.invariant.

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special relativity david berman queen mary college university of london

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total distance

distance y2

distance x2

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distance y1

spacetimethe distance

distance unchanged

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