We are now ready to expand our 3-D space into 4 Dimensions.  My ability to draw diagrams in 4-D is very limited, so we will not have any this time.

We need to pick a point in 4 dimensions that does not exist anywhere in our 3-D world.  At the start, we might be tempted to think that such a world is simply unphysical and impossible until we consider that the 4^th dimension might be related to time.  Consider a point (x, y, z, t), where ‘t’ represents the time (watch) that is associated with the point (x, y, z).  Does this even make sense, points running around with timer watches?

     It might be easier to believe if we consider a single point that is able to move around in 3-D space.  When we track the same point (x, y, z) over time, we can imagine that it doesn’t have to be in the same position all the time relative to where it is right now.  If we want to track the movement of a point through time, then its associated timer watch (t) comes in handy.  We can think of the path of the point walking through time as walking through a path in 4-D “space-time”, where the point (x, y, z, t) describes the position of the point (x, y, z) at the time ‘t’.

Suppose that we choose the origin (0, 0, 0, 0) to be the position of the point when we set the timer to zero.  Then a “new point” in 4-D space-time could be (0, 0, 0, 1) which describes the same point in the same place 1 second later. We are getting closer to constructing our new 4-D space.  Now we need to draw line segments from every point in 3-D space to this new point in 4-D space-time and then need to locate the shortest of these line segments (like we did when expanding 2 dimensions to 3 dimensions).

A problem arises, however, because the time ‘t’ is not a distance, so it is confusing to find a measure for distance in this 4-D space.  What is x² + y² + z² + t²?  One way to solve this problem is to somehow convert the time ‘t’ into a measure of distance:

    (Distance/Time)*Time = Distance

By choosing a constant speed ‘C’  (with units of Distance/Time) that is the same for every point in this 4-D space, we have a way to convert any time into a distance:

      C*t = the distance traveled in time ‘t’

So, if C is truly a constant speed for all points in space-time, then ‘C*t’ is the distance that any point with this constant speed ‘C’ would travel in a time ‘t’.  So let us redefine a point in 4-D space time to be described by:

   (x, y, z, C*t)

And we have a 4-D space of distances.  Then distance ‘S’ between the origin (0, 0, 0, 0) and any point (x, y, z, C*t) is found to be:

   S² = x² + y² + z² + (C*t)²

And, as usual, the direction ‘D’ of a line from the origin to any point can be expressed as the vector:

    D = (D_x, D_y, D_z, D_t)

      where

    D^.D = D_x² + D_y² + D_z² + D_t² = 1

Here D_t expresses how fast time is changing as you travel a 4-D distance ‘S’ along the line.  As usual, lines from the origin in 4-D spacetime can be expressed as:

     (x, y, z, C*t) = (D_x, D_y, D_z, D_t)*S

                        or

     (x, y, z, C*t) = (D_x*S, D_y*S, D_z*S, D_t*S)

‘D_x*S’ expresses how far the point has traveled in the x-direction, as usual, and the other 3-D coordinates have similar meanings.  The new one is ‘Dt*S’ that describes how much time has elapsed after we have traveled a distance ‘S’ along this 4-D line.  D_t determines the speed of the timer.  Notice that the 4-D distance ‘S’ is not the same as the 3-D distance we will now call ‘S₃’:

     S₃² = (D_x*S)² + (D_y*S)² + (D_z*S)²
         = (D_x² + D_y² + D_z²)*S²

Since 1 – D_t² = D_x² + D_y² + D_z², we see a more interesting expression for S₃:

   S₃² = (1 – D_t²)*S²

This is interesting, and maybe a bit surprising, because it says that the distance that the point (x, y, z, C*t) travels in 3-D space is completely determined by the D_t direction parameter of the 4-D line.  If you look at it a bit closer, however, it starts to make sense.  Since 1 – D_t² = D_x² + D_y² + D_z² then when D_t = 1, the other direction parameters, D_x², D_y², and D_z² must add to zero.  Thus (D_x, D_y, D_z, D_t) must be (0, 0, 0, 1), and the point is only moving in the ‘time’ direction, but not moving in any of the 3-D directions.  This 4-D direction describes a point that is not moving in 3-D space, S₃² = 0.

If D_t is less than 1 then the point has room to move in 3-D space.  A more perplexing situation happens when D_t = 0.  Then in this direction, time stands still, no time is elapsing, but the point can go any distance it wants in the 3-D space.  Physically this makes very little sense.  The point could be everywhere all at once. Let’s look at what this means in terms of the speed ‘V’ of the point.  Speed is the 3-D distance it travels per the time elapsed ‘t’ (recall C*t = D_t*S):

    V = S₃/t
    V = C*S₃/(D_t*S)

    V² = C²*S₃²/(D_t*S)²
         = C²*(1 – D_t²)*S²/(D_t²*S²)
         = C²*(1 – D_t²)/D_t²

And we see that the speed of the point is constant everywhere on the 4-D line and it is completely determined by how fast the point is traveling in the ‘time’ direction, D_t.  Things also start to be more understandable.  However, if D_t goes to zero, then the speed is infinitely fast (you can’t divide by zero).  Having things that can go at infinite speed is very problematic.  The point could literally be everywhere at once.  One fun definition of time is: “Time is something that keeps everything from happening all at once”.  It makes sense to have some sort of a speed limit for our points.

Einstein, a brilliant physicist, made the argument that the velocity ‘C’ (a constant speed for all points in space-time) should be that speed limit.  If we assume this (C² >= V²), then we can find restrictions on the direction parameter D_t:

     C² >= C²*(1 – D_t²)/D_t²
      1 >= (1 – D_t²)/D_t²
      D_t² >= ½

This puts some serious restrictions on the directions of lines that are possible to traverse in space-time:

    ½ <= D_t² <= 1

D_t is between the square root of ½ and 1.  Another restriction that we see is that we cannot go backwards in time, D_t cannot cross over to become negative unless the point violates the speed limit and can go infinitely fast.

So, a 4-D space-time can exist mathematically, but for it to make sense physically, certain restrictions must be placed on the directions of lines we can draw in this space-time. Next time, we will look at how to find (D_x, D_y, D_z, D_t) in space-time and their associated angles.  We will introduce a new angle, A_t, that describes the angle of the “time” direction.  We will find that D_t = Sin(A_t).  At the speed limit ‘C’, D_t is the square root of ½.  From trig this means that A_t = 45 degrees.  The restrictions require that A_t is between 45 degrees and 90 degrees.  This “cone” of 4-D space-time is the only place where anything can exist when it comes from the origin.  It is commonly called the “light cone”. And the speed of light is the speed limit ‘C’.