Invariant Spacetime Interval in Minkowski Diagram

This section provides a demonstration of the invariant spacetime interval property using a Minkowski diagram.

Another property of the Minkowski spacetime model is that the interval of two events is a constant in all intertial frames. This is called invariant spacetime interval.

The invariant spacetime interval property can be demonstrated by a simple thought experiment described below:

• Amy stays in frame moving at a speed of v=0.6*c relative to Bob's frame.
• Amy observes an event E occurring at (X',cT')=(6,2) in her frame.
• Amy derives that the interval E and the origin O in her frame is sqrt(-(cT')**2+(X')**2) = sqrt(-4+36) = sqrt(32).

Now in Bob's frame:

• The event E would be observed at (X,cT)=(9,7) in Bob's frame based on the Minkowski diagram given below.
• Bob derives that the interval E and the origin O in his frame is sqrt(-(cT)**2+(X)**2) = sqrt(-49+81) = sqrt(32).

So the interval between events E and O is a constant in both frames.

Of course, the invariant spacetime interval can be derived from the Lorentz transformation:

```  X' = gamma*(      X - beta*c*T)       #3: Lorentz Transformation
c*T' = gamma*(-beta*X +      c*T)       #4: Lorentz Transformation
gamma = 1/sqrt(1-beta**2)               #7: "gamma" factor
beta  = v/c                             #8: "beta" factor

s = sqrt(-(c*T)**2+X**2)               #13: Interval: (X,cT) & (0,0)
s' = sqrt(-(c*T')**2+X'**2)            #14: Interval: (X',cT') & (0,0)

s' = sqrt(-(gamma*(-beta*X+c*T))**2+(gamma*(X-beta*c*T))**2)
#15: merging #3 and #4 into #6

s' = sqrt(gamme**2*(-(-beta*X+c*T)**2+(X-beta*c*T)**2))
s' = sqrt((-(-beta*X+c*T)**2+(X-beta*c*T)**2)/(1-beta**2)

s' = sqrt((-((beta*X)**2 - 2*beta*X*c*T + (c*T)**2))
+(X**2 - 2*X*beta*c*T + (beta*c*T)**2)))/(1-beta**2))

s' = sqrt(-(beta*X)**2 - (c*T)**2 + X**2 + (beta*c*T)**2)/(1-beta**2))
s' = sqrt((1-beta**2)*X**2 - (1-beta**2)*(c*T)**2)/(1-beta**2)
s' = sqrt(-(c*T)**2+X**2)

s' = s                                 #16: Interval is a constant
```

Last update: 2014.