The purpose here is explain a bit about the geometry of Minkowski space and to show why "proper time" defined in terms of the "spacetime interval" has anything to do with what clocks measure.

This explanation is based on a geometric treatment of special relativity. It is largely based on the treatment given in the book The Geometry of Minkowski Space time by Gregory L. Naber. You can refer to that book for more detailed mathematics and to Essential Relativity, Special General and Cosmological by Wolfgang Rindler for a less mathematical and more physical treatment. Some of Rindler's perspective is also a part of this note.

This argument carries over unchanged to general relativity simply by a localization observation -- the metric of GR is locally the Minkowski metric.

The bottom line is that proper time is the only time that is measured by any (ideal) clock. It is the time that is associated with the world line of the clock, and is intimately related to the arc length of that world line and thereby to the geometry of spacetime.

What is Minkowski spacetime ?

Minkowski spacetime is the setting for special relativity. It is by definition ordinary 4-space with a non-degenerate quadratic form of signature (+,-,-,-). Equivalently one can use a quadratic form of signature (-,+,+,+) and this is the convention used by Naber, but we will use the other convention here.

The quadratic form defines an inner product on Minlkowski space. It is analogous to the dot product of ordinary Euclidean space, except that it is not positive-definite. This means that It is possible for the inner product (squared length) of a vector with itself to be negative , and it is possible for the inner product of a vector with itself to be zero even if the vector is not the zero vector. So, in Minkowski space a nonn-zero vector can be perpendicular to itself. That fact requires you to readjust your intuition with regard to some geometric ideas, so don't get blindsided by some of this weirdness.

Now just as with the ordinary inner product, there is no a priori need to define a basis so as to express it as a "dot product". To take the ordinary product of two vectors in 2-space just form the product of their lengths and the cosine of the angle between them – no basis needed to do this geometrically. So think of the Minkowski inner product as a geometric idea, and we'll talk about the relationship to a basis set.

It takes a little more work than in the usual case of a positive-definite inner product, but one can show that given a non-degenerate inner product one can find an orthonormal basis for the underlying vector space. In this more general case an orthonormal basis is a basis in which distinct elements are orthogonal (have inner product 0) and in which the inner product of any basis element with itself is 1 or -1. One can prove and any two orthonormal basis sets always have the same number of elements with inner product with themselves equal to -1, and that defines the "signature" of the quadratic form. In the case of Minkowski space the signature is (+,-,-,-). The inner product of a vector with itself is called the "squared norm". A vector with a negative squared norm is called "space-like" and one with a positive squared norm is called "time-like".

We will denote the inner product, using the Minkowski quadratic form, of 4-vectors X and Y by <X,Y> and then then length of a vector X is the norm of X,

Transformations that preserve the inner product are (inhomogenous) Lorentz transforms, sometimes called Poincare transforms. One generally restricts attention to a subset of the full set of Lorentz transforms for physical reasons, but that is a subject for another time. Lorentz transforms correspond to individual observers and serve to relate coordinate measurement for one observer to another observer. Objects that are preserved by Lorrentz transforms are called invariants of special relativity.

For the purposes herein we will work in units in which the speed of light is 1. That makes the usual formula for gamma simply

Length in Minkowski space The length of a vector X is just |X|. The length of a curve is given in the usual way. A parameterized curve in Minkowski space is just a function from the real line, or a line segment taking as its values 4-vectors in Minkowski space. Let be such as curve, defined on [0,1]. Then the length of (arc length)is just

as in the case of ordinary Eudlidean space with a positive definite inner product, one can define an arc length parameter for φ [ ref

(http://www.math.hmc....es_and_arc.html)], call it s by

One can parameterize a curve using arc length, and one finds then that the "speed" along the curve is simply 1.

Proper Time in Minkowski Space

The proper time separating the end points of a curve in Minkowski space is simply the length of the curve, and the proper time parameter,, is simply arc length. This is simply a definition.

The obvious question is what the definition of "proper time" ,, has to do with "time", t, since t (in special relativity) is what is measured by the clock of an observer in the rest frame of the observer, and on the surface is nothing but distance associated with an unconventional notion of "length". So far we have worked purely in terms of mathematics and the geometry of Minkowski space. To address this new question requires physical reasoning.

Consider a curve in Minkowski space that consists of short displacements in space at constant speed. Any smooth curve is approximable by such a curve. This curve represents the trajectory of a particle in Minkowski space, and in the reference frame of that particle we select an orthonormal basis x,y,z,t.

Now consider one increment of displacement, from to where the displacement is timelike the length of the displacement is just

=

=

=

Or

This shows that is the time sensed by a clock that is co-moving with the particle on this small segment. Since any smooth curve is approximated by a series of such segments it follows that the arc length along the curve is identifiable with the time experienced by a particle moving along the cure. So the parameter is deserving of the term "time".

Since your wristwatch, if you wear it, follows your world line, your proper time is precisely what is recorded on your (ideal) wristwatch. For that reason proper time is sometimes colorfully called "wristwatch time" for pedagogical reasons.

In general relativity one loses the notion of a reference frame, except locally, but the notion of proper time remains just as in the simpler case of special relativity. That is simply because in general relativity we deal with a Lorentzian 4-manifold, wherein each point is contained in a neighborhood that is locally Minkowski space. Thus all of the reasoning above remains in force, since arc length is defined locally in terms of the local metric and simply "patched together" using the atlas and differentiable structure of the manifold. Proper time is what clocks measure, and that is the definition of time.