Spacetime intervals?

Question

Why is the spacetime interval -(ct)^2 +x^2+y^2+z^2=s^2 ? Also how do we prove that it is invariant from every reference frame?

OK well it's oviously invariant, but WHY is that the definition. Why is there a -c^2t^2 term. why that exact term?

Answer 1

The interval (which can be defined as you do it, or more commonly today as minus what you put) is invariant under a Lorentz transformation (defined as a combination of boosts and ordinary rotations).

It should be manifestly obvious that it is invariant under a simple rotation.

Give your coordinate system a boost in an arbitrary direction (which dillates the time and contracts the length in the boost direction--choose z for simplicity). Recalculate the interval with the new coordinates and show that it is invariant.

Answer 2

Let see how the earth moves in terms of space and time..
One dislpacement of the earth is equal to the space volume wihch was displaced.The cycle continues however the amount of space volume dispacement is a function of time . So the more time goes by the more volume the movement of the earth has displaced and it leaves a trail behind.So space displacements and time are related.
Since the Sun losee mass and the Earth gained mass their mass
time displacement are either increasing or decreasing. so in the real physical world spacetime is not invariant.

In the world of math we can asociated space in various dimensions. Einstein associated it with 4 dimensions. In which he associated the parameters to the pyhagorean triangle whose solution is two fold ;one positive real and the other negative imaginary. However the pythagorean hypothenus of a triangle is only an arbritrary invariant and so the the space time invariant.

String theory the lastest theory of the Universe which relates space time and volumes exist are formulated into 11 dimensions. . So Einstein-Minkowsky space time concept of 4 dimensions has be replaced with 11 dimensions. and in other theories it goes up to 26 dimensions.
So Minkowsky's world of space time has become out dated.