A motivation for the minus sign in the Minkowski metric

A motivation for the minus sign in the Minkowski metric Antonio J. Pan-Collantes October 13, 2025 Abstract This short note provides an intuitive motivation for the signature of the Minkowski metric, specifically the relative minus sign between the time and space components. By drawing a direct analogy with the familiar Euclidean metric in a 2D plane, we show how the invariant quantity in each space naturally determines the form of its metric. In the Euclidean case, the invariant is the length of a vector, preserved under rotations. In the Minkowski case, the invariant is the proper time between two events, preserved under Lorentz boosts. This comparison demonstrates why the geometry of spacetime is pseudo-Riemannian rather than Riemannian. Introduction One of the first and most fundamental points of confusion when studying special relativity is the origin of the minus sign in the Minkowski metric. Why is the geometry of spacetime described by dτ2=dt2 −dx2 −dy2 −dz2, and not, for instance, by a purely Euclidean metric with all plus signs? This is a classical source of doubt for many students, and the author has not, so far, found a completely satisfying motivation for this specific signature in standard textbooks. The purpose of this short note is to provide an intuitive motivation for what the Minkowski metric is and why it must have that signature. We will do this by drawing a direct analogy with the more familiar Euclidean metric. To keep the comparison as clear as possible, we will restrict our analysis to two spatial dimensions in the Euclidean case, and one spatial and one temporal dimension in the case of Minkowski spacetime. Also, we will work in an infinitesimal neighborhood of the origin, as if the point or events we will talk about were infinitesimally near the origin. 1 The Euclidean plane Suppose we are beings living in a 2D plane, like ants on a tabletop. We can construct coordinates to label the points in our world by fixing a point Oand drawing perpendicular axes, as usual. We will call our coordinates xand y. We will restrict ourselves to this kind of system of coordinates (with Ofixed), because they are physically equivalent (there is no natural choice of a preferred vertical axis, there is no preferred direction). By basic geometry, the relation between two of these coordinate systems is given by a rotation of angle θ: x′ y′=cos θ−sin θ sin θcos θx y. Consider a stick lying on the tabletop, with one of its endpoints in O. Any two ants, using different coordinate systems (x, y) and (x′, y′), will disagree on the coordinates of the other endpoint. But, importantly, they will agree on the coordinate y′′ assigned by a third ant whose y-axis is aligned with the stick. This is an invariant, and we can define it as the length of the stick. See Figure 1. Then, any other ant can compute the length of the stick in his own coordinates by transforming the coordinates of the endpoint in her own system into the coordinates in the system of the ant aligned with the stick, and taking the y′′ coordinate. I.e., 0 y′′=cos ϕ−sin ϕ sin ϕcos ϕx y.(1) 1 x y xP yP x′ y′ x′ P y′ P x′′ y′′ stick O Figure 1: A stick viewed from different coordinate systems. Observers using the black (x, y) or blue (x′, y′) systems will measure different coordinates for the stick’s endpoint. But both observers will agree on the value of y′′ coordinate read by a third observer using the green system (x′′, y′′ ), which is rotated to align its y′′-axis with the stick. From (1) we obtain two equations, and we can eliminate ϕto get l2:= y′′2=x2+y2. This is the key step to measure lengths in a 2D pure space world. 2 The Minkowski Plane Now, suppose we are beings living in a 1D world, but in addition, we can also measure time. We can label the points of our spacetime by fixing a point Oand drawing a time axis tand a space axis x. The point Orepresents the origin of space and the origin of time. But we have different equivalent systems for the labeling. In this case, the equivalence is not due to an arbitrary choice of a vertical axis, but to the fact that there is no preferred state of rest. Any observer moving with constant velocity can consider herself at rest, and would yield different labels for the events. A crucial turning point in physics was the experimental discovery that the speed of light in a vacuum is the same for all inertial observers, regardless of the motion of the light source. This counter-intuitive fact, famously supported by experiments like the one conducted by Michelson and Morley, stands as a fundamental postulate of special relativity. As is standardly derived in the literature [4], the only linear transformations that preserve this invariant speed are the Lorentz transformations. For a boost along the x-axis, the coordinate transformation between two observers takes the form of a Lorentz boost: x′ t′=cosh θ−sinh θ −sinh θcosh θx t. Now, the notion of proper time enters the scene. Consider two facts that coincide in time and space for one observer. Then they must agree on space and time for all observers. Therefore, if we consider, for example, a spaceship travelling from Earth to Alpha Centauri, and if the crew observes that their clock is at 0 when they are leaving and at 3 years when they are arriving, any observer must agree in the fact that, with respect to the time measurement system of the ship, the travel has had a duration of 3 years. This quantity is an invariant, and it is called the proper time. For more on this argument, see [6]. So now, we can define the proper time between two events as the time measured by an observer for whom the two events happen at the same point in space (note the analogy with the length measurement in the Euclidean case). See Figure 2. Any other observer can compute the proper time in her own system by transforming the coordinates of the events into the system of the observer for whom the events happen at the same point in space, 2 x t xP tP Event Worldline O t′ x′ t′′ x′′ Figure 2: An event in Minkowski spacetime viewed from different inertial frames. Observers using the black (t, x) or blue (t′, x′) systems will measure different time and space coordinates for the event’s occurrence. However, they will all agree on the time, t′′ , measured by an observer (green system (t′′, x′′)) for whom the event happens at the same spatial location. The t′′-axis of this comoving observer is defined as the proper time. and taking the t′′ coordinate: 0 t′′=cosh ϕ−sinh ϕ −sinh ϕcosh ϕx t. From here, we have two equations, and we can eliminate ϕto get τ2:= t′′2=t2 −x2. 3 Conclusion The minus sign in the time interval expression is a consequence of the constancy of the speed of light, which gives rise to Lorentz boosts as the natural change of coordinates, and the physical significance of the notion of proper time. Some references, like [5, 6], use a graphical representation called Epstein diagrams and the idea of everything moving at the speed of light, in a kind of spacetime parametrized by (τ, x), to justify the presence of the minus sign. But for the author of this paper, this is a misleading approach. On the one hand, it is not justified why we can consider every object moves at light speed. Indeed, it seems that this can be concluded once we know the appropriate expression for the time interval. And on the other hand, τis not an appropriate coordinate, since it depends on the path. References [1] A. Einstein, ”Zur Elektrodynamik bewegter K¨orper” (On the Electrodynamics of Moving Bodies), Annalen der Physik, 322(10): 891–921, 1905. [2] H. Minkowski, ”Raum und Zeit” (Space and Time), Address delivered at the 80th Assembly of German Natural Scientists and Physicians, Cologne, 21 September 1908. [3] E. F. Taylor and J. A. Wheeler, Spacetime Physics, W. H. Freeman, 1992. [4] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, 1973. [5] L. C. Epstein, Relativity Visualized, Insight Press, 1985. [6] FloatHeadPhysics, ”Why the Metric has a Minus Sign”, YouTube, 20 Oct. 2023, https://www. youtube.com/watch?v=TcOLyqfA5k8. 3