“What are the Units of Proper Time \(d\tau\) in Special Relativity?”
This post discusses “proper time” as defined by the special relativity theorists. Briefly we are not persuaded that this quantity denoted \(d\tau\) represents a measure of “time” at all. The definition of proper time is a Lorentzian idea which involves a difference of squares. But the strict Riemannian geometer must understand that “differences of squares” do not represent “sums of squares”. Indeed how does a Riemannian geometer talk about “distances which satisfy reverse triangle inequalities” with a straight face? This implies that the Lorentzian “proper time” is not a measure of “time” at all, but a heterogeneous quantity relating Lorentz to Riemannian lengths. This is cause of much confusion in the literature.
It’s typically said that proper time is the “self time” as measured by an observer’s clock in the observer’s inertial reference frame. And this is a strange expression. We have discussed elsewhere “What is Time?” that “time is matter in motion” ex definitio. Therefore a “self time” is any periodic motion with a counter (wrist watch, pendulum, dog wagging it’s tail, comet appearing and disappearing, sun rising and setting). There is no assumption of uniformity in the motion, although for practical purposes this is useful. But special relativity deliberately undermines the possibility of “objective uniformity”. Clocks and measuring sticks are flexible and wobbly in the SR worldview (which view we do not share).
Lorentz Lengths versus Riemannian Lengths
Consider the Lorentz Minkowski line element \[g=ds^2=-c^2dt^2+dx^2+dy^2+dz^2.\] In our critical analysis of special relativity we have warned readers from naively applying Riemannian ideas to Lorentzian metrics. We repeat: Lorentzian metrics are not Riemannian! When a Riemannian geometer starts working with Lorentz metrics, they typically begin with the line element \(ds':=\sqrt{-ds^2}\) on timelike curves \(\gamma\) where \(ds<0\). By habit they refer to integrals \(\int_\gamma ds'\) as “lengths” along the curves \(\gamma\). And they assume that this Lorentz length describes a Riemannian length. But this interpretation is misleading since Lorentz “lengths” satisfy the reverse triangle inequality, and thus “Lorentz lengths” are antithetical to “Riemannian length” This is trivial observation and repeatedly overlooked. Again we insist the measures represented by “Lorentz length” expressions \(\int_\gamma ds'\) are not “lengths” as understood by Riemannian geometry. Countless errors are introduced by confusing these two definitions.
Related to the Lorentz metric is the so-called “proper time” function on timelike curves, usually represented by a definition type formula \(ds'=c d\tau\). But we find this “definition” insufficient and not strictly defining a “proper time”. Why? Because \(d\tau\) does not have units of time. This is directly related to the error of interpreting \(ds'\) as a measure of “Riemannian length”. In otherwords, \(d\tau\) formally represents a ratio of “Lorentz length” over “Riemannian length” multiplied by “time”, i.e. \[[d\tau] = \frac{[\text{Lorentz length}]}{[\text{Riemannian length}]} [\text{time}].\]
Sums of Squares versus Differences of Squares: Pythagoras and Anti-Pythagoras
As strange as it first sounds, we believe it’s a significant mistake to believe \(d\tau\) has physical units of time. The fundamental problem is again that \(ds'\) is emphatically not a measure of metric distance. Indeed the Pythagorean theorem involves sum of squares and and is inherently Riemannian. But Lorentz lengths are represented by differences of squares, and there is no “Pythagoras” in this setting. In euclidean geometry, the squareroot of a sum of orthogonal squares represents a length because of Pythagorean identity. There are right angled triangles to be constructed whose hypotenuses are the lengths in question. However the squareroot of a sum of signed squares is not readily identified as a length to the Pythagorean, even if one assumes the squares are orthogonal. There are no right angled triangles to draw except on the null cone \(ds^2=0\) where the Pythagorean \[c^2 dt^2=dx^2+dy^2+dz^2\] is equivalently the hypothesis on the velocity of \(c\) in vacuum. But on the null cone the proper time \(d\tau\) is identically zero.
Brief Remark on Diagonalized Rest Frames
\(\newcommand{\del}{\partial}\)
Here we include a frequent argument in the general relativity, alleging to establish the identity \(ds'=c d\tau\) where \(d\tau\) represents the so-called “proper time” of an inertial observer as measured by their local “clock”. This interpretation comes from the use of a so-called “instantaneous rest frame”. This requires the observer to find coordinates \((\tau, \xi, \eta, \zeta)\) where \(\tau\) represents “time” and all the partial derivatives vanish: \[\frac{\del \xi}{\del \tau}=\frac{\del \eta}{\del \tau}=\frac{\del \zeta}{\del \tau}=0.\] In this particular coordinate system one finds \(ds^2=-c^2d\tau^2\) and \(ds'=\sqrt{-ds^2}=c d\tau\).
But can we really conclude that \(ds'\) has units of \([length]\) as a general rule, or equivalently that \(ds'/c\) has units of \([time]\), based on a computation in one coordinate system? We observe that the system of equations is not tensorial since the partial derivatives \(\del \xi/ \del \tau\) are not covariant. However the vanishing of the covariant derivatives is an invariant quantity, i.e. invariant with respect to change of variable rather than covariant (contravariant) with respect to change of variables. Replacing the partial derivatives with covariant derivatives, the vanishing no longer implies the coordinates \(\tau, \xi, \eta, \zeta\) can be integrated into the “diagonal” form for \(g\). Therefore we are not really persuaded by this argument’s identification of “proper time”, and nonetheless the derivation still has the inherent problem of not representing “proper time” in units of “time”.
Carets and Non Tensors
The “mere coordinate singularities”, so named by Misner-Thorne-Wheeler in their influential textbook on “Gravitation” leads to their introduction of “geometrodynamics” and “carets” in the GR. An introduction to these ideas are discussed in (Klauber 2001) and briefly in (Thorne, Wheeler, and Misner 2000). The caret argument says: although the components of tensors are noninvariant, the projections of these tensors onto coordinate components in a “local Lorentz frame” are well defined, observable and measurable. Thus carets are the coordinates of tensors in the so-called local Lorentz frame.
This idea of taking projections in a local Lorentz frame is curious, and though perhaps intuitively appealing, the mathematics is not clear. For fixed spatial coordinates \(\hat{x},\hat{y},\hat{z}\) we know from Riemannian geometry how to construct the projections of an arbitrary vector \(v\) onto the basis vectors, i.e. \(\hat{x}\cdot v\). But there is no projection onto the time variable, i.e. \(t\) and \(\hat{t}\) has no meaning. There really is no objective length or magnitude or “unit” of time.
In practice MTW (Thorne, Wheeler, and Misner 2000) take projections by dividing and renormalizing “component wise” in the tensors, but we claim this is ad hoc procedure and only convenient when the metric is diagonal. The same argument is encountered in (Klauber 2001). Most mathematicians/physicists are not persuaded by these critiques. The projection onto coordinates axes seems to be inspired by projections in quantum mechanics. In the Born-Dirac Copenhagen interpretation, the projections amount to “observations”. The carets of MTW attempt to develop a parallel idea in GR.
Many will disagree here, but it depends on “Who taught you GR and how were you taught?” There appears to be no large-scale consensus anyway, and we welcome comments to the contrary.
[-JHM]