General Relativity

Special and General Relativity

Einstein developed two theories of Relativity :-

1. Special Relativity - Explains relative uniform motion between two reference frames.

2. General Relativity - Explains relative non-uniform motion ( taking accelerations due to Gravity into account ) between two reference frames.

Special Relativity has been extensively discussed within the Higher Physics course ( see Special Relativity ) and as such, the Advanced Higher course focuses on General Relativity.

General Relativity

General Relativity is Einstein's explanation as to "How Gravity works". It is based upon the concept that Mass distorts the space around it (more specifically the Space-Time around it), and that Gravity is a consequence of this curvature of space. The diagram below sums up General Relativity in its simplest terms as described by John Archibald Wheeler :-

"Matter tells Space-Time how to curve; Space-Time tells Matter how to move"

Matter tells Space-Time how to curve :-

Space-Time tells Matter how to move :-

Note - The above diagrams demonstrate the curvature of Space-Time using the rubber-sheet analogy. This shows a 2-dimensional representation (a flat Sheet) of the 4-dimensional Space-Time.

The explanation of movement within Space-Time explains why the path of a ray of light is distorted by passing near a Mass. A ray of light (and any other object) will follow the shortest possible path (known as a Geodesic) through curved Space-Time. This causes the ray of light to follow a curved path and from the point of view of an external observer, the path of the light bends.

The Equivalence Principle

In the National 5 course the acceleration due to Gravity (g) was given two different units, either ms-2 or Nkg-1 . The first Unit (ms-2) states that the Object is accelerating, whilst the second Unit (Nkg-1) states that the object is experiencing a Gravitational Force. Throughout the National 5 course, these two concepts were seen as interchangeable (and they are) , but no reason as to why was given. The two units (and therefore concepts) are interchangeable due to the understanding of Einstein's Equivalence Principle.

The Equivalence Principle states that :-

"No experiment can differentiate between the effect of a Uniform Gravitational field and an accelerating reference frame."

The above statement means that an experiment performed from a stationary position within a Uniform Gravitational field will give the same results as an experiment performed in the absence of a Gravitational field inside an accelerating region ( i.e. a spacecraft ) :-

The Equivalence Principle can be demonstrated by the two Spacecraft above. In a windowless Spacecraft, it is impossible to distinguish between a Spacecraft accelerating at 9.8 ms-2 and a Spacecraft stationary at the Earth's Surface. The Reaction Force experienced within the accelerating Craft mimics the Gravitational Force on the Earth's surface exactly.

The Equivalence Principle can also be demonstrated with a Projectile thrown horizontally within the two Spacecraft. In both cases, the Projectile will follow a curved path, in the Spacecraft's Reference Frame.

Diagram One - the Projectile ( once thrown ) does not experience the acceleration of the Craft, and travels in a Straight Line ( in the projectile's reference frame ) , but due to the relative acceleration of the Craft, the path is Curved ( in the Craft's reference frame ).

Diagram Two - The Projectile experiences an acceleration vertically due to the Gravitational Field, and as such the path is Curved.

The same effect is also seen with Light. The Path of Light in an Accelerating Reference Frame is Curved due to the Relative acceleration, and therefore in a Gravitational Field the path of a beam of Light Curves.

Gravitational Lensing

A consequence of Curved Space-Time is that Starlight passing the Sun will follow a curved path, and as such will appear in a different position within the Sky :-

This apparent shift in Stellar Position was used to "prove" Einstein's Theory of General Relativity. In 1919, Sir Arthur Eddington observed the shift in Position of Stars in the Constellation Taurus during a Solar Eclipse to match exactly the predictions of General Relativity.

Identical effects can be seen on the Scale of Galaxies, except in this case, its not a single Star that is distorted, but the emitted light of a whole Galaxy :-

Image result for gravitational lensing

Depending on the relative sizes and position of the two Galaxies, an Optical Effect known as an Einstein Ring can occur. An Einstein Ring is a two-dimensional consequence of Gravitational Lensing and can be seen in the image below :-

In the above image, the central yellow region is a nearby Elliptical Galaxy. The ring around is the Einstein Ring, the Optical distortion of a more distant Galaxy.

Note - Gravitational Lensing is being used to identify regions of high Dark Matter density. Dark Matter causes light to experience Gravitational Lensing, and in the absence of a visible Object to cause the effect, Dark Matter can be located.

The video below shows a summary of Gravitational Lensing :-

Gravitational Redshift

Einstein's Theory of General Relativity can also be used to explain the concept of Gravitational Redshift. The Diagram below shows the visual effect on a Ray of ( Initially ) Blue Light emitted by a Massive Object :-

As the Photon moves away from the Mass, Work must be done to escape the Gravity Well. As was seen in the Higher Physics course, the Energy of a Photon is related to Frequency by :-

As stated above, Work must be done by the Photon. This Work Done constitutes Energy "lost" by the Photon. If the Energy of each Photon is reduced, the Frequency of the Photon is also reduced and therefore the longer the Wavelength of the Photon ( Redshift ) .

The amount of Work required related proportionally to the Mass of the Object, therefore the More Massive an Object, the greater the Redshift of the Light.

This has been shown experimentally ( The Pound-Rebka Experiment ) using Gamma Rays within moving Lifts.

The Video below shows a summary of Gravitational Redshift and the Pound-Rebka experiment :-

Gravitational Time Dilation

The above Observation of Gravitational Redshift can also be explained through the concept of Gravitational Time Dilation. If the vibrations of a Wave are used to measure time, as the Wave experiences Redshift the period of the Wave increases. This implies that as the Gravitational Field increases, the slower Time Passes. The diagram below shows an example of this, with the dotted line representing the Schwarzschild Radius of the Mass.

As the Mass is compressed and the Gravitational Field increases, the Wave becomes "stretched". If each Wave Peak passing a point is used to measure time, then the Time becomes Dilated. The limit of this case is at the Event Horizon, at which point the Wave becomes infinitely stretched, giving a flat line, and therefore no emitted Wave. This means that a Clock moving towards a Black hole will appear to an outside Observer to run slower and slower until the Clock stops ticking at all at the Event Horizon.

If this process is ran in the opposite direction, for an Observer within a Gravitational Field observing a clock "Further out", the Clock will be Observed to run Faster. This can be Physically Observed in the GPS Satellite System, where due to the Satellite's position "further out" in the Earth's Gravitational Field, the on-board clocks "run fast" compared to clocks on Earth.

Note - In the Higher Course, taking into account Special Relativity gave the clocks on the GPS Satellites as running Slower not Faster. In fact both conditions are true :-

GPS Clocks run slower by ~ 7 μs per day due to time dilation following Special Relativity ( Moving clocks run slow )

GPS Clocks run faster by ~ 45 μs per day due time dilation following General Relativity ( "Higher up" Clocks run faster )

The Net Effect is that the Clocks on the GPS satellites run fast by ~ 38 μs per day.

What is Space-Time ?

The fundamental explanation of General Relativity revolves around the concept of

Space-Time, but so far in this section it hasn't been explicitly defined. Space-Time is a 4-dimensional coordinate system which takes into account that an event will occur at a location with a set 3-dimensional position ( x,y,z ) and time ( t ) . This can be represented graphically using the following four-quadrant graph :-

Note - The Origin of this graph shows the Observer's Present Location and Time. Also, the above shows the Space-Time Graph for object moving in only one of the 3 Spatial Dimensions ( the x-direction ) .

The Diagram below shows a graphical representation of three objects in Space-Time, relative to the x-direction :-

The above diagram shows how an object can move within Space-Time. Each coloured line represents the path taken by an object ( Think of it like the maps in Indiana Jones ) and is referred to as the Object's World Line.

However, as shown in other sections of the Physics course, the Speed of Light places an upper limit on velocity, and as such there are velocities which are not allowed :-

In the above diagram, Objects a, b and c are travelling at speeds which Obey the Laws of Physics, whereas d and e are travelling at speeds exceeding the Speed of Light. This means that an Object's World Line cannot cross ( from left to right ) the Line denoting the World Line of Light.

However, World Lines can cross from right to left. In this case, the crossing point denotes when an Event is Observable. The diagram below shows an example of this :-

In the above diagram, a stationary Observer at point P will not be aware of an Event that took place at the Origin until time t0 , as information travelling at the Speed of Light will take this amount of time to reach Point P.

The diagram below shows a 2-dimensional representation the Space-Time Graph showing an Event occurring at the Origin :-

In the above diagram, the Event is only Observable within the Light Cone. Any observer outwith the Light Cone cannot Observe the Event as they lie too distant for information to been transferred to that point in Space and Time.