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The Relativity Theory's Implications for Mathematical Finance ... Relativity, relativistic volatility, invariant uncertainty-time, from world economy to universe economy, stochastic velocity dependent volatility. 1 Introduction ...... dose of gamma rays.
Space-Time Interval
Consider two events that are close together in space-time. Just as we can compute the distance between two points in space, dl, we can also compute the distance between two nearby events in space-time. The important fact about the space-time distance, called the space-time interval, is that it is invariant; that is, it is measured to be the same by all observers and its value does not depend on how we choose to label the events.
The space-time interval ds is defined by
The difference dt is called the coordinate time difference to remind us of the fact that its value depends on how we choose to label the events, in contrast to the interval ds which does not.
Notice something peculiar about the interval. Its square can be positive, zero or negative!
If ds2 < 0 the two nearby events are said to separated by a time-like interval.
If ds2> 0 the two nearby events are said to be separated by a space-like interval. If ds2 = 0 the two events are said to be separated by a light-like interval.
The time measured at any given point is called the proper time. The difference dt is the proper time that has elapsed between the two events. It is the time measured by a clock at that point. The elapsed proper time is just |ds2|1/2/c.
Now consider setting dt = 0. In this case
Consider two events that are close together in space-time. Just as we can compute the distance between two points in space, dl, we can also compute the distance between two nearby events in space-time. The important fact about the space-time distance, called the space-time interval, is that it is invariant; that is, it is measured to be the same by all observers and its value does not depend on how we choose to label the events.
The space-time interval ds is defined by
- ds2 = -(cdt)2 + dl2
The difference dt is called the coordinate time difference to remind us of the fact that its value depends on how we choose to label the events, in contrast to the interval ds which does not.
Notice something peculiar about the interval. Its square can be positive, zero or negative!
If ds2 < 0 the two nearby events are said to separated by a time-like interval.
If ds2> 0 the two nearby events are said to be separated by a space-like interval. If ds2 = 0 the two events are said to be separated by a light-like interval.
Proper Time and Proper Distance
- ds2 = -(cdt)2
The time measured at any given point is called the proper time. The difference dt is the proper time that has elapsed between the two events. It is the time measured by a clock at that point. The elapsed proper time is just |ds2|1/2/c.
Now consider setting dt = 0. In this case
- ds2 = dl2
- (a) ds2 = -(cdt)2 + dl2
- dt = |ds2|1/2/c = dt[1 - (v/c)2]1/2.
dt = |ds2|1/2/c is the expression for the proper time difference not only for observers at rest in the coordinate system but for all observers, however they move,provided that they move only along time-like world-lines.
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