Minkowski Metric

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At the center of the theory of relativity is the Minkowski metric. If you understand what this formula says, then you are a long way in understanding the structure of space-time.

Distance in space

Distances in space are given by the following metric:

x^2+ y^2 = r^2

On a sheet of paper with x- and y-axes, you can calculate the distances between points with this formula. It is the Pythagorean Theorem.

All the points that are at the same distances away from a given point lie on a circle!

Distance in space-time

Now, on a ‘sheet of space-time paper’ with an x-axis and a ‘t-axis’, you can calculate distances between points (which are ‘events’) with the following formula:

Delta s^2 = Delta t^2 - Delta x^2

With this formula you can calculate the ‘distance’ between events. Yes, just like there is a ‘space’ distance between China and America, there is a ‘spacetime’ distance between the first landing on the moon and the breaking  of the Berlin wall. In words it says:

text{Spacetime distance} ^2 = text{temporal distance}^2 - text{space distance}^2

Where the distances are between events. Just like a point in space is a location, a point in ‘spacetime’ is an event.

All the points that are at the same distances away from a given point lie on a circle!

All the points that are at the same distances away from a given event lie on a hyperbola!

The difference between space-time metric and space metric:

The minus sign results in that a bigger temporal distance between events results in a smaller space-time distance between this events! This is in contrast with distances in space:

A bigger y-component distance between points means automatically a bigger space-distance between points!

Why does it make more sense that an event 10 km away from you and next year has a ‘smaller distance’ away from you than an event 10 km away and next day? Well, the difference is very small, but the reason that the event next year is ‘closer by’ is because

But why?

What does ‘spacetime distance’ mean? Does it make any sense? Is it of any use of knowing the ‘spacetime’ distance?

It stands for the time experienced between the events for an inertial frame where the two events happen at the same spot. It is the maximal amount of time someone can experience between the events. This is also called the ‘proper time’.

The proper time is usually denoted by a ‘tau’-symbol. So if we rewrite and rearrange the minkowski metric we get the following formula:

Or in other words:

Delta tau^2 + Delta r^2 = Delta t^2

Now it looks more like a proper Pythagorean metric that we are familiar in everyday life. It also looks like the circle-formula:

x^2+ y^2 = r^2

Impliciations of this metric:

Even though it doesn’t seem like it, mathematically time is perpendicular to space. This has geometric implications which we have found in nature. Actually, we first found the implications and deduced the metric from them. The implications are:

  1. Speed of light is invariant with respect to observers
  2. Time dilates in inertial frames with respect to you
  3. Length contracts in inertial frames with respect to you
  4. Energy has inertia
  5. Velocity add in a complicated matter
  6. You can calculate distances between events, which are invariant.

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