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Special relativity

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The theory of special relativity was published by Albert Einstein in 1905. The theory deals with a special case where the effects of gravity can be ignored hence it is called "special." Einstein later incorporated gravitation into the theory of General Relativity.

Postulates

Special Relativity has two postulates from which it is developed.

  1. Observation of physical phenomena are the same for all inertial observers
  2. The speed of light in vacuum (c) is the same to all inertial observers. It is the same in all directions regardless of the velocity of the source or the observer.

Space Time Diagram

Minkowski diagram.png

This is a space time diagram; also called a Minkowski diagram it shows the relationship between inertial reference frames. Such that if observer S' is moving with a speed v in relation to S. The resulting superposition has the axes of S' tilted such that the slope of the time axis is v / c and the slope of the space axis is c / v.

Length Contraction

Srlc1.png

Relativistic length contraction is a foreshortening of the length of an object in the direction of relative motion, as illustrated above.

Srlc2.png

  • Formula for relativistic length contraction.
  • v is the relative velocity.
  • c is the speed of light.
  • L is the rest length.
  • L' is length at velocity v.

Std1.png

  • Space time diagram showing relativistic length contraction.

An examination a space time diagram shows this affect is caused by a tilting relationship between two reference frames in space-time. The yellow area is a light cone. The black line are the space (x) and time (t) axis of . The blue line is the time axis of frame S'. The purple line is the space axis of a ship in frame S'. Note that in frame S' the light cone reaches both ends of the ship at the time while, they don't in frame S. That because when observer S is looking at both ends of the ship is looking a different points in the frame S' time lined. So the forward end of ship S' as seen by observer S has not moved as far as the aft end, this results in ship S' being foreshortened to observer S.

It needs to be noted that an observer in frame S' will see objects in frame S with the same length contraction.

Time Dilation

Dt1.png

  • Light Clock at rest.

The concept of time dilation is based largely on a clock with a pulse of light going from a emitter, (red) to a receiver. (blue) the trip time is a unit of time.

Dt2.png

  • Light Clock in motion.

When the clock is in motion the light pulse has to travel further to reach the detector, and because the speed of light is constant it take more time for the light pulse to reach the detector. This translates to the formula for time dilation do to relative motion.

Dtf.png

  • v is the relative velocity.
  • c is the speed of light.
  • t is the rest length.
  • t' is length at velocity v.

Now this one again works both ways, the results in what is called the twin paradox. However it is simple to illustrates why the twin taking the trip would experience less time.

Dts0.png

Consider a space ship going from Earth to the nearest star system; 4.45 light years away; at 0.866 c. The above image shows the the ship with its 0.5 length contraction. To an observer on Earth the trip will take 5.14 years, producing a round trip time of 10.28 years.

Dtsv.png

However to the ship's crew the stars and the distance between them will be shortened in the direction of motion to 0.5 of what is observed on Earth, resulting in a travel distance of only 2.23 light years and a corresponding 2.57 year travel time. This produces a round trip time of only 5.14 years. Thus the ship's crew would experience less time than those on Earth.

If one of the astronauts on the ship had a twin that stayed on Earth. he would return home to find his brother about 5 years older than himself.

Now this takes care of the physical affect of time dilation but it does not reconcile how each twin would observe the other during the trip. The solution to this observational problem can be found in the relativistic Doppler effect. It includes the affects of both time dilation and distance, hence we can calculate just how fast each twin will observe time to flow in the other's reference frame.

Earth to ship. ____________________ Ship to Earth.
  • The back line is Earth's path through space time.
  • The dark blue line is ship's path outward path through space time.
  • Purple line is its path return path through space time.
  • The green dots are equally spaced points in time within their respective frame of reference.
  • The red lines are the path of light between Earth and the ship during the outward trip.
  • Light blue dots arrival point of light.

On the outward trip the relativistic Doppler effect shows each twin would see 1 second pass for the other twin for every 3.7 seconds of his own time. The twin on the ship would see this effect for 2.57 years, as such he would see his twin age 0.69 years. However since the light showing the ships arrival would not arrive at Earth for 4.45 years, the twin on Earth would see this effect for 9.59 years, as such he would see his twin age 2.57 years.


Earth to ship. ____________________ Ship to Earth.
  • Blue lines are the path of light between Earth and the ship during the return trip.
  • Light blue dots arrival point of light.
  • Left image shows Earth to ship.
  • Right image shows ship to Earth.

On the inward trip the relativistic Doppler effect shows each twin would see 3.7 seconds pass for the other twin for every second of his own time. The twin on the ship would see this effect for 2.57 years, as such he would see his twin age 9.59 years, for a total of 10.28 years. However since the light showing the ships departure would not arrive at Earth for 4.45 years, the twin on Earth would see this effect for only 0.69 years since ship returns after 10.28 years, as such he would see his twin age 2.57 years, for a total of 5.14 years.

Earth to ship. ____________________ Ship to Earth.
  • This diagram shows the light path's for both legs of the flight.
  • Left image shows Earth to ship.
  • Right image shows ship to Earth.
  • Note that there is more blue on the left and more red on the right. That is why the ships crew sees the Earth age more than they do.

Thus the twin paradox is resolved both in terms of the physical affect of time dilation and the observation of both twins, since both the physical affect and observation agree on how much each twin would age.

Relativistic Mass

Rmf.png

  • Formula for relativistic mass.
  • Formula 1

It from this that we get Einstein famous equation for the equivalence of mass and energy. E = mc2

Ke1.png

  • Formula for relativistic kinetic energy.
  • Formula 2

It turns out that when v << c that the classical formula Kec.png is a good approximation of relativistic kinetic energy. So the relativistic increase in mass with velocity is a result of the object's kinetic energy.

Biblical Significance

Special Relativity has been criticized by some Christians. It is claimed that by removing absolute space and time, it is there by implying no moral absolutes. First of all this is illogical since there is no real connection between the two. Furthermore Special Relativity does have an absolute, the speed of light.

1 John 1:5 compares God to light, and according to Special Relativity no matter how fast you accelerate you will always fall equally short of the speed of light. Similarly, no matter how hard you work you will always fall equally short of the glory of God.

Special Relativity shows that time is an integral part of the structure of the universe, and not just the perceptions of a sequence of events. The result is that anything outside the universe would be timeless (or have its own time). God, as the creator of the universe, must exist outside His creation, and therefore by definition must be eternal and transcendant.

References