Proper Time
"I am convinced that He ( God ) does not play dice with the universe"  Albert Einstein ( 1879  1955 )
Nature's "Sphere of Light" is mathematically known by the light sphere equation:
This is true for any relatively moving system to stationary system where , speed of light, is a finite universal constant, however measured in whatever coordinate system is under investigation; and all of the above is made true by the Lorentz Transformation Equations ( FitzGerald  Lorentz Contraction Equations )
§ The Meaning of Proper Time:
It is the time "attached" to a particle or body in the most extreme existential sense of the light carrying information as to the relative motion of the particle or body to some measuring instrument.
§ Proper Time Defined:
§ Newer meaning for "distance":
The old idea of distance as being
Hence, in the pseudo  Euclidean "Minkowski World" all the original vector relations are preserved in a generalized and powerfully elegant manner!
§ Proof that the Proper Spacetime Interval ( or Proper Time ) is Lorentz Invariant:
That is, this spacetime interval unlike those for time and distance which vary for different observers in relative motion, is invariant or the same for all observers across all frames of reference provided that a Lorentz coordinate system is used  i.e., that Lorentz transformation laws are valid for whatever coordinate system is employed. See: Glossary definition for Spacetime Interval.
^{∗} A better expression for the speed of light combining time and space is 1 meter of light travelled in 1/299,792,458 seconds.
§ Corollary: 2nd Proof that the Proper Spacetime Interval ( or Proper Time ) is Lorentz Invariant:
That is, Lorentz invariance of proper time also demands that the Proper Time Interval is Lorentz invariant!
§ Meaning of the Lorentz invariance of the Proper Time Interval:
The Principle of Relativity ( Galilean  Newton or Einstein Special Relativity ) states that the laws of physics are invariant in every inertial ( free  floating ) reference frame. Therefore, while different observers in overlapping relative, free  floating, frame motion to each other will assign different space  time coordinates to a the same event, , the Principle of Relativity demands that the Lorentz invariant proper space  time interval possesses the same value as calculated by all observers in every overlapping free  floating frame of reference, but measured directly only at the "attached time" of the particle, body or on the wristwatch of a person in common sense terms.
Pythagorean Theorem Hidden in Lorentz Invariant Proper Time
Combining systems and , we get
§ Example:^{∗∗}
An event occurs in a relatively moving frame of reference such as a flash of light which is also viewed in a stationary frame of reference.
^{∗∗}note : this example is used in the future upcoming Relativity Science Calculator Mac application for Proper Time.
Proper Time and Energy Revealed
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