Minkowski Spacetime

Minkowski spacetime provides a lucid pictorial representation for the special theory of relativity. An event occurring at a time t at the location (x,y,z) in three-dimensional space is described by a point ((Subscript[x, 0]),(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3]))≡(c t,x,y,z) in a four-dimensional manifold known as Minkowski spacetime.The factor c=2.9979×(10^8) m/s, the speed of light, gives (Subscript[x, 0])=c t the dimensions of length, to match those of (Subscript[x, 1]), (Subscript[x, 2]), (Subscript[x, 3]). The fundamental principle of special relativity can be expressed as the invariance of the interval d(s^2)=d((Subscript[x, 0])^2)-d((Subscript[x, 1])^2)-d((Subscript[x, 2])^2)- d((Subscript[x, 3])^(2 ))as measured by observers in all inertial frames. This differs dramatically from Galilean relativity, the foundational postulate of Newtonian mechanics, in that both time and space intervals become relative, or dependent on the velocity of the observer.

For simplicity, this Demonstration considers just a single space dimension x that, along with the time variable c t, gives a two-dimensional projection of Minkowski space. A stationary observer measures variables denoted x and c t, while an observer in another inertial frame moving at a constant speed v measures variables denoted x' and c t'. For ease of visualization, the primed variables are shown in red. These alternative variables are related by a Lorentz transformation, x'=x cosh θ-c t sinh θ and c t'=-x sinh θ+c t cosh θ. Here θ is the rapidity, defined by θ=arctanh(v/c). The velocity can also be expressed in dimensionless form as β=v/c. In accordance with this hyperbolic transformation, the primed (red) coordinate axes are skewed at angles θ relative to the stationary axes. The origin represents an event in which x=c t=0 momentarily for both observers. You can move the locator to determine another event, for which the two observers no longer agree on values of x and c t. The two sets of values of x and c t are represented by the black and red projections on their respective axes. Their numerical values are given at the bottom of the graphic. The constancy of the interval is shown by the relation ((Δs)^2)=(((c t))^2)-(x^2)=(((c t'))^2)-(((x'))^2). In the cases ((Δs)^2)>0, ((Δs)^2)<0, and ((Δs)^2)=0, the interval is called timelike, spacelike, or lightlike (or null), respectively. Timelike intervals lie within the future or past lightcones, projected as yellow triangles in the graphic. The red lines meeting at the event point are parallel to their respective red axes. Note that time is not ordered in a spacelike event: past and future are not invariant; nor is space ordered in a timelike event: left and right are not invariant.

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Snapshot 1: This spacelike interval could represent the length of a rod. The stationary observer measures a length that is shorter than that measured by the moving observer. This is called the Lorentz-FitzGerald contraction, given by L=(Subscript[L, 0])(Sqrt[1-(v^2)/(c^2)]).

Snapshot 2: The interval here could represent time measured in the moving frame. The black and red axes might interchange to represent the moving and stationary observers, respectively (it's all relative!). The time interval in the moving frame, t=0.740/c , represents the proper time. The longer time measured by the moving observer shows time dilation.

Snapshot 3: The event marked by the locator lies in the future lightcone. This means that this event could possibly be caused by an event at the origin.

Snapshot 4: The event lies in the past lightcone, meaning that it might possibly be the cause of the event at the origin.

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Contributed by: S. M. Blinder

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