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What Is Time in Spacetime?

It is well-known that the hyperbolic plane is "too big" to be embedded in Euclidean 3-space, but it is not "too big" to be embedded in Minkowski 3-space. Whereas the metric of the former is positive definite, the metric of the latter is indefinite. Hence. the latter seems a "natural" to distinguish between space and time by their differing signs in the metric. However, rather than being an attribute, the Minkowski metric highlights the fallacy of associating time with the coordinate that has a sign change in the hyperbolic sheet.


If s is "proper" time and t is coordinate, or lab, time, then special relativity tells us they are related by

ds^2 = dt^2 - dr^2 - r^2 d\phi^2 (1)

where r is the radial coordinate and \phi the azimuthal angle in units where c=1. General relativity "generalizes" the line element (1) by adding "gravitational potentials" A and B,

 ds^2 = A dt^2 - B dr^2 - r^2 d\phi^2 (2)

where in the "static" scenario A and B can only be functions of r. In the most interesting case of the Schwarzschild metric, AB=1.


Now consider the upper sheet of a hyperboloid in Minkowski 3-space. The hyperboloid sheet is

T^2 -X^2 -Y^2 =1 T>0 (3)

Since the corresponding metric is

ds^2 =-dT^2 +dX^2 + dY^2 (4)

it makes it a "natural" to try to associate T with time in (1) and the X and Y with the space coordinates. This, will, in fact, be the Achilles' heel of relativity.


First, let us observe that metric (4) can be parameterized by two "angles", \alpha and \beta. With

T= cosh\alpha

X = sinh\alpha cos\beta

Y= sinh\alpha sin \beta,

the indefinite metric (4) is transformed into the definite metric

ds^2 = d\alpha^2 + sinh^2\alpha d\beta^2. (5)

Expression (4) clearly points to the fact that the distinction between space and time has completely disappeared!


Whereas (1) is "flat", the embedded metric (5) has Ricci curvature, R=-2. We might try to rearrange (1) into an Euclidean metric,

dt^2 = ds^2 + dr^2 + r^2 d\phi^2 (6)

following Epstein, where dt is the Euclidean increment in "length" and the increment in proper time, ds, becomes the polar angle in the Epstein chart. But, what about the angle \phi in (6)?


Actually, what is being considered is the metric (4) on the hyperbolic plane. Writing

\rho = \sqrt{X^2-T^2} (7)

and

\phi=tanh^(-1)(T/X) (8)

the metric (4) becomes

ds^2 = d\rho^2 + dY^2 - rho^2 d\phi^2. (9)

Thus, the time increment dT is given by a polar angle \phi.


The "fly" in the ointment is (8). For we can write it as

\phi = (1/2) ln[(X/T+1)/(X/T-1)] (10)

which would make \phi a hyperbolic velocity where it not for the fact that the Euclidean velocity X/T>1, making the particle a tachyon. If we really wanted to make relativistic sense out of this we would replace (7) and (8) by

\rho = \sqrt{T^2-X^2} (11)

\phi = tanh^(-1)(X/T) (12)

For then we would obtain the same metric (9) but it would correspond to the Minkowski line element

ds^2= dT^2 -dX^2 + dY^2 (13)

simply because we switched T and X in (7) and (8). But it is the change of variables (11) and (12) that make relativistic "sense" since X/T<1. although it is nonsensical to associate and angle with a hyperbolic measure of the velocity.


Consequently, one cannot associate the variable T with time in the hyperboloid sheet (3). This makes all spacetime diagrams usied in general relativity completely meaningless.



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