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SPACE AND TIME: Part 1C

PART 1C: Minkowski's words are in **boldface**.

HYPERBOLOID OF TWO SHEETS

Minkowski has told us that, for Newton,
space and time can be united, but only as a fibre bundle without a unique
time-direction. He now begins the investigation which will lead to special relativity:
he looks for a "metric structure", in effect a geometry, of four-dimensional
space which *depends on some parameter,* call it *c*,
in such a way that for some limiting case, the time axis loses its uniqueness and the Newtonian
fibre-bundle is recovered. If such a parametrised geometry exists,
perhaps it, and not the fibre bundle, describes the real world.

**"Now what connection has the condition of orthogonality in
space with this complete upward freedom of the
time axis? To exhibit the connection we take a positive
parameter c and consider the locus
**

**
It consists of two sheets separated
by t=0 analogous to a hyperboloid of two sheets."**

In the graphic above, *ct* is vertical, and *z*
is not shown.

**"Considering the sheet in the region t > 0 we now conceive those
homogeneous linear transformations of x, y, z, t into
four new variables x′, y′, z′, t′,
in which the expression for this sheet of the hyperboloid in the new
variables corresponds to the original expression.
Evidently the rotations of space about the origin belong to these
transformations."**

In the figure above, no axes have been drawn, although
the vertical *t*-direction and the the horizontal plane are
obvious. One could put *x* and *y* axes *anywhere*
in the horizontal plane, and the object depicted would still be
a two-sheeted hyperboloid oriented along the vertical axis. Its
equation, therefore, according to basic analytic-geometry,
must be *c² t² - x² - y²* = 1. If one were
to choose some other pair of right-angled axes *x′*
and *y′*, the same would be true, only with
primes:
*c² t² - x′² - y′²* = 1.
Rotating the space axes leaves the equation of the hyperboloid
unchanged.

If one tilts the *t*-axis away from the vertical,
the situation is different: the
hyperboloid is *leaning* relative to the vertical.
This lean cannot be eliminated by simply
rotating the space axes. However, maybe there is some *other*
kind of transformation, not a rotation, which one can perform on the
space-coordinates to get a new system *x′, y′, t′*
which will formally look the same as the unprimed equation of the
hyperboloid.

**"We shall next obtain a full understanding of the remaining transformations
by considering one in which y and z remain unchanged."**

That is, we will consider only the *x* and *t* coordinates,
so we can draw the figure easily on paper: instead of
an hyperboloid or, worse, its four-dimensional
generalisation, we can draw a simple hyperbola. We will work here with
the figure from the printed version of Minkowski's talk.

**"Let us draw
the intersection of this **[upper]**
sheet with the plane of the x- and
t-axes, the upper branch of the hyperbola
c² t² - x² = 1 with its asymptotes."**

**"Then let any radius vector OA′ of this branch of the
hyperbola be constructed from the origin O."**

**"Let the tangent to the hyperbola at A′ be extended
to the right until it intersects the asymptote at B′."**

**"Let OA′B′ be completed to form the parallelogram
OA′B′C′,
and finally for later developments let B′C′ be continued to
D′, its intersection with the x-axis."**

C′ is the intersection of a line through the origin parallel to the tangent A′B′ with a line through B′ parallel to OA′.

**"If we then take OC′ and OA′ as axes for parallel
coordinates x′ and t′ with
units OC′=1, OA′=1/c, then this branch of
the hyperbola again has the equation
**

**
and the transition from x, y, z, t to x′, y′,
z′, t′ is of the type under consideration."**

To which most readers might be forgiven for responding: "Oh is it now?" Minkowski offers no proof; perhaps to him it was obvious. In the next installment, we will fill in the gap. Readers whose algebra is rusty may wish first to review the idea of a unit vector, discussed in the Khan Academy video below.

TIME AND SPACE, by Hermann Minkowski