Introduction to Einstein's Theory of Special Relativity

1. Einstein's Theory of Special Relativity is a Theory of Relationship between Space and Time.
   
   This theory is based on the experimentally validated postulate that Speed of Light (i.e Electromagnetic Waves) in Vaccum ( denoted by letter \(c\) ) has a Constant Value of 299,792,458 m/s, irrespective of the Speed/Velocity of the Source of the Light and/or the Speed/Velocity of the Observer. This is the Maximum Permisible Speed of Any Object in Universe and No Other Object in Universe can ever achieve an Absolute or Relative Speed Greater Than or Equal to this Speed.
2. As a result of the constraint on the Maximum Permisible Speed on Every Object in Universe (which must be \( < c\) ), the Space and Time instead of being independent of each other, act as a Single Continuous Entity known as the Space-Time Continuum.
   
   An Event in Space-Time Continuum is a Snapshot of an Occurance at Given Location at a Given Time.
   
   An Observer of Events (or an Object) is the Frame of Reference with respect to which measurements are done in Space-Time Continuum. All Observers ( Frames of References ) are Always in Motion, either in Space or in Time or Both.
   
   A Static Observer (Static Frame of Reference) is an Observer which Does not Change its Spacial Loaction with Time. Hence it's motion in Space-Time Continuum is only in Time.
   
   Any Event occurring in the Space-Time Continuum or Any Observer of Events are not only tracked with respect to their Location in Space (given by 3 Spacial Coordinates \(x\), \(y\) and \(z\)), but also with with respect to their Time of Occurance/Existance (given by Time Coordinate \(t\)). Thus, Every Event occurring in the Space-Time Continuum and Every Observer of Events is given by their corresponding Space-Time Coordinates (\(x\), \(y\), \(z\), \(t\)) ( or (\(t\), \(x\), \(y\), \(z\)) ) or corresponding Homogeneous Space-Time Coordinates (\(x\), \(y\), \(z\), \(ct\)) ( or (\(ct\), \(x\), \(y\), \(z\)) ). The Homogeneous Space-Time Coordinates have all their values in the Units of Length / Distance.
3. Also, because of the constraint on the Maximum Permisible Speed on Every Object in Universe, the Spacial Distance between Observer/Event \(A\) having Space-Time Coordiate (\(x_1\), \(y_1\), \(z_1\), \(t_1\)) and Observer/Event \(B\) having Space-Time Coordiate (\(x_2\), \(y_2\), \(z_2\), \(t_2\)) Must be Lesser than the Distance Light can Travel in the Difference between time \(t_1\) and \(t_2\). That is,
   
   \(\sqrt{{(x_1-x_2)}^2 + {(y_1-y_2)}^2 + {(z_1-z_2)}^2} < c(t_1-t_2)\)   ...(1)
   
   Setting \(\Delta x = x_1-x_2\), \(\Delta y = y_1-y_2\), \(\Delta z = z_1-z_2\), \(\Delta t = t_1-t_2\) and Squaring Both Sides of equation (1) above we get
   
   \({\Delta x}^2 + {\Delta y}^2 + {\Delta z}^2 < c^2{\Delta t}^2\)
   
   \(\Rightarrow {\Delta x}^2 + {\Delta y}^2 + {\Delta z}^2 - c^2{\Delta t}^2 < 0\)   ...(2)
   
   \(\Rightarrow c^2{\Delta t}^2 - {\Delta x}^2 - {\Delta y}^2 - {\Delta z}^2 > 0\)   ...(3)
   
   \(\Rightarrow {\Large \frac{{\Delta x}^2}{c^2}} + {\Large \frac{{\Delta c}^2}{c^2}} + {\Large \frac{{\Delta z}^2}{c^2}} - {\Delta t}^2 < 0\)   ...(4)
   
   \(\Rightarrow {\Delta t}^2 - {\Large \frac{{\Delta x}^2}{c^2}} - {\Large \frac{{\Delta c}^2}{c^2}} - {\Large \frac{{\Delta z}^2}{c^2}} > 0\)   ...(5)
   
   The expressions given by Left Hand Side of equations (2), (3), (4) and (5) represent the Space Time Interval which describes the Net Separation Between Any 2 Events (or an Event and an Observer or 2 Observers) in Space-Time Continuum.
   
   The Value of Space Time Interval between Any 2 Given Events is Constant (i.e. Invariant) across Different Observers (Frames of Reference).
   
   Also, the Value of Space Time Interval can be calculated in terms of units of Distance (as given in equation (2) and (3) above) or in terms of units of Time (as given in equation (4) and (5) above).
4. Having a constraint on the Maximum Permisible Speed on Every Object in Universe has following effects on Perception of Events occurring in the Same Location in Space and/or at the Same Time by Different Observers.
   1. Events that take place at Same Time in Different Locations appear to happen at Different Times when obseverved by Different Observers.
   2. Events that take place at Same Location at Different Times appear to happen at Different Locations when obseverved by Different Observers.
   The 2 phenomena as given above are referred as Relativity of Simultaneity.
   
   In addition, Following Abberations are Observed by any Observer moving with a Speed/Velocity Near to the Speed of Light
   1. Time Dialation: Difference in Percieved Time of a given Event for Different Observers. For any Given Observer, Events appear to happen Slower for Other Observers than they actually are.
   2. Distance Dialation: Difference in Percieved Distance Between 2 Events at a Given Time for Different Observers. For any Given Observer, Distances Between 2 Events appear more for the Other Observers than they actually are.
   3. Length Contraction: Difference in Percieved Distance between 2 Events at a Given Time for an Observer. Distances between 2 Different Events Appear Shorter than they actually are.
   4. Duration Contraction: Difference in Duration between 2 Events at a Given Position for an Observer. Duration between 2 Different Events Appear Shorter than they actually are.