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Einstein's Relativity

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Einstein's Relativity - Interesting Facts

  • Matter determines how space curves.   Curved space determines how matter moves.

  • Due to the natural curvature of space, the shortest path between any two objects is never a straight line, but a curved line called a geodesic.   An example of this is that we can see stars that are located in a straight line behind the sun appearing near the edge of the sun.   This occurs because the strong gravitational effect of the sun curves space in such a way that the shortest distance for light to travel is the geodesic path curved around the edge of the sun.   An earth-based analogy is that aeroplane flight paths follow geodesic paths instead of straight lines around the curved earth surface to save both time and fuel.   Bernhard Riemann defined space as a 'topological manifold of an arbitrary number of dimensions' in 1854, unlike our perception of space in 3 (straight) dimensions.   This geometry is a fundamental part of Einstein's general theory of relativity.

  • One second is exactly 9,192,631,770 beats of a Cesium atom, very close to 1/(24 x 60 x 60) of an 'earth rotation' day.   An earth rotation day is the varying time it takes for the earth to rotate once relative to the sun.

  • The speed of light in a vacuum is exactly 299,792,458 metres per second.   Light reaches all objects from all directions at the same speed, regardless of their motion.   Even if you travel at 240,000 metres/second, light approaches you head-on at the same speed it reaches you from behind.   So, regardless of the speed of light sources and receivers, light always travels at the same speed.   It also follows that light has no concept of time, because (to light) all distances are zero, and therefore it would perceive that it reaches its destination instantaneously.   It was this thinking which led Einstein to his theories of relativity.

  • The above 2 exact definitions of a second and of the speed of light define the exact length of a one metre: it is the distance travelled by light in a second, divided by the speed of light in metres per second.

  • No physical object can travel at or faster than the speed of light.   The speed of light is generally considered to be a physical speed barrier.

  • The age of the universe is widely believed to be 12 to 13 billion years old, and still expanding as a result of the 'big bang'.   This produces a type of horizon in space, where light has not yet reached Earth from objects further away than 12 to 13 billion light years.  

  • * If two clocks are separated by a large distance, different observers will disagree about any time difference between them.   Some will say the clocks indicate the same time, others will say one clock is ahead of the other, and still others will say the opposite.   Also, different observers will disagree about whether the clocks are running normally, or faster, or slower than normal.   But all will agree that the two clocks are running at the same speed.

  • * If two clocks are moving with respect to each other and pass nearby each other, all observers will agree on what each clock indicated when they passed.   But different observers will disagree about whether the clocks are running at the same speed or whether one is running faster (and which one).

  • * Consider the following: A large number of space-buoys is setup in straight line each separated by a large distance.   You start out in a rocket near the central bouy.   While you are at rest with respect to the buoys, you observe that the clocks on all the buoys indicate the same time (i.e. they are all synchronised).   You rapidly accelerate along the line of the buoys.   You will observe 3 sudden changes in the buoys:
    1. The spacing between them has been reduced,
    2. The clocks on the buoys are all running slower than normal,
    3. The buoys ahead of you have jumped ahead in time while those behind you have jumped backward in time; the farther away the buoy, the greater the jump.

  • The Doppler effect causes objects moving away to have their light spectrum red-shifted while objects approaching have their light blue-shifted.   This really means that the wavelengths of light they radiate (or reflect) are moved downward or upward on the frequency spectrum.   These measurements were the first clue that the universe is expanding.

    However, it does not mean that visible light is more red (or more blue in the unfortunate event of a fast approaching object).   Visible light is only a small portion of radiation; there also exists significant infra-red (longer wavelengths below visible red) and ultra-violet (shorter wavelengths above visible violet) radiation.

    White light is the equal combination of all wavelengths of visible light through all colours of the rainbow from red to violet.   In the case of red-shift:

    • what was previously red becomes invisible infra-red
    • some colours may remain as different visible colours with a longer wavelength (eg blue may become yellow)
    • some of what was previously invisible ultra-violet becomes a visible colour
    so we still continue to see a full visible colour spectrum as white.

Formulas
  • Gravity is the curvature of time-space.   Like velocity, it also affects the "rate of time".   Einstein's special theory of relativity was published in 1905, and is the theory demonstrated in the example above.   In 1915, Einstein published his general theory of relativity which included the effects of gravitation and acceleration.   Einstein (1879-1955) is more famous for relating energy to mass in his formula:
    E = mc2
    His greatest achievement, however, is in discovering the true nature of our universe by way of pure imagination!   His own quote is "Imagination is more important than knowledge".

  • As objects approach the speed of light, their "rate of time" (calculated by others) approaches zero, and distances (travelled by the object) approach zero, and their mass (as measured by an observer) increases.   Here's the formula for the 'relativity factor' for an object where c is the speed of light.   It is important to realise that the relativity factor is the factor by which:
    • a stationery object calculates the rate of time slowed in the moving object
    • a moving object calculates distances contracted (in the direction of travel, less at other angles to zero in directions perpedicular to the direction of travel)
    • each object calculates an increase in mass of the other
    Rf = (1-(s^2)/(c^2))^0.5
    where s is the speed of the object, and c is the speed of light.

  • If 2 spaceships travelled towards each other, each travelling at 240,000 km/sec (80% the speed of light relative to a stationery object), at the point the pass each other, they would not measure each other's speed as 480,000 km/sec (which would be faster than the speed of light).   In this case, each would correctly measure the other's speed to be nearly 293,000 km/sec (or nearly 98% the speed of light).
    Relative Speed = (Sa + Sb) / (1 + (Sa x Sb / c2) )
    At this crossover speed, they would each measure the length of the other ship as just under 20% of its length at rest, and also slightly rotated due the curvature of space between them.

  • * Suppose we observe an object move a differential amount through the space-time continuum. It moves dx in the x-direction, and dy & dz in the y- and z-directions.   We observe that it ages by da and that this happens during the interval dt on our own clocks.   Now let us consider the quantity:
    dS = Sqrt( (dx)2 + (dy)2 + (dz)2 + c(da)2 )
     This is the differential displacement through space-time.   Thus the quantity dS/dt is the speed at which the object is moving through space-time.

    You may be surprised to learn that the following equation is always true: dS/dt = c.   Thus all objects are always moving through space-time at c.   When we observe that an object is at rest in space, it is aging at the most rapid possible speed (da = dt).   On the other hand, if an object is moving at c through space (e.g. dx/dt=c) then da/dt = 0.   Thus photons (and anything else travelling at c) won't age.

    We cannot make anything move faster than c because we cannot change the speed of anything through space-time.   Everything moves at c.   We can change the direction (to mostly through space (dx/dt large), or entirely through time (dx=0, dy=0, dz=0)), but we can never change the speed.   When objects are at rest in space, they are moving at maximum speed through time and when they are at rest in time, they are moving at maximum speed through space.

  • Here's a formula that takes into account both the relativity factor and Doppler effect for an object approaching us at speed v:
    Sqrt((1 + v/c)/(1 - v/c))

    The shift is the inverse for a receding object, or:

    Sqrt((1 - v/c)/(1 + v/c))

* Many thanks to Paul Alan Cardinale for providing the "Relativistic Tidbits" marked with an asterisk.

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