*Notes to a video lecture on UNIZOR.COM*

__Past and Future__

in Minkowski Space+Time

in Minkowski Space+Time

We continue analyzing the movements restricted to a two-dimensional

*XY*-plane for visual representation of the three-dimensional Minkowski

*XYT*-coordinates

Assume, a point at present time

*is located at the origin of*

**t=0***XY*-plane.

As it moves along any trajectory (

*) on a plane, the world point that represents this movement will be at position (*

**X(T),Y(T)***) in the three-dimensional Minkowski space+time.*

**X(T),Y(T),T**The distance from the world point (

*) to the*

**X,Y,T***T*-axis equals to

**D(T) = √X²(T)+Y²(T)**As we know, the speed of movement cannot exceed a speed of light in vacuum

*.*

**c**Therefore, during time

*the point on the*

**T***XY*-plane cannot deviate in its movement from the origin by a distance greater than

*and the world point (*

**c·T***) cannot deviate from the*

**X,Y,T***T*-axis by the same distance

*.*

**c·T**From this follows that for any time

*the following inequality is held*

**T***or*

**d(T) ≤ c·T**

**X²(T) + Y²(T) ≤ (c·T)²**The equation

**X² + Y² = (c·T)²**represents a conical surface in three-dimensional

*XYT*-space.

Therefore, the inequality

**X²(T) + Y²(T) ≤ (c·T)²**represents the

**inside of this cone**.

Below is a graphical representation of this with indication that if our present time is

*, the inside of an upper cone represents all possible points reachable in the future, while the inside of a lower cone represents all possible points in the past where the moving point could have been located in the past.*

**T=0**If a point in the past (for negative time

*) was located outside the lower cone, there is no way it could reach the origin of coordinates at current time*

**T***, because its speed would have to exceed the speed of light*

**T=0***.*

**c**Analogously, any point in the future (for positive time

*) outside of an upper cone could not be reached from the origin during the time*

**T***, because its speed would have to be greater than the speed of light*

**T***.*

**c**Therefore, we can state that all

**world lines**that represent all the past and future possible movements of an object located at the origin of coordinates at time

*must lie inside the cone described by an equation*

**T=0**

**X² + Y² = (c·T)²**This cone is called the

**light cone**because its shape (actually, the angle at the top) explicitly depends on the speed of light

*.*

**c**Of course,

__any position on a world line__can be interpreted as the beginning of a motion with all prior positions along a trajectory considered as the

**past**and all subsequent as the

**future**.

In this case we can require that at any position the world line should be contained within a

**light cone**constructed at it, which can be schematically illustrated by this picture.

Extending the definition of the

**light cone**from three-dimensional Minkowski

*XYT*-frame (that we used for better graphical representation) to a practical four-dimensional Minkowski

*XYZT*-frame (that, unfortunately, we cannot visually represent), we can define a similar

**light cone**as

**X² + Y² + Z² = (c·T)²**Now let's express the same thing in term of metrics, assuming the distance between two

**world points**in four-dimensional Minkowski

*XYZT*-frame, presented in the previous lecture

*Space+Time*, is expressed as

**D²(A,B) =**

= c²·(T= c²·(T

_{B}**−T**_{A}**)² − (X**_{B}**−X**_{A}**)² −**

− (Y− (Y

_{B}**−Y**_{A}**)² − (Z**_{B}**−Z**_{A}**)²**The condition for two

**world points**to be inside of a

**light cone**(that is, for them to represent the state of some object at two different moments of time, thus belonging to its

**world line**) can now be expressed as

**the square of a distance between them in Minkowski space+time, as defined in the above formula, to be**, which makes the distance itself to be a positive real number.

__non-negative__If the square of a distance between two points in the Minkowski space+time is negative and, consequently, the distance itself is imaginary complex number, these two points cannot belong to a world line that represents a motion of one point-object, because, to reach one point from another, this object would have to move with a speed faster than a speed of light in vacuum, which is impossible.