A Lorentzian metric distinguishes clocks, light cones, and volume

At each event, a Lorentzian metric has one negative and three positive directions in an orthonormal frame. For our convention a nonzero timelike tangent has negative squared norm, a null tangent zero norm, and a spacelike tangent positive norm. The signature cannot change under a smooth invertible coordinate transformation. A metric becoming genuinely degenerate inside a valid chart is therefore not simply a switch between coordinate conventions. Metric components can still carry different units when some coordinates are lengths and others dimensionless angles.

Along a future-directed timelike worldline, the proper-time element is the positive square root of minus the line element divided by c. Integrating it requires the actual path, not only its endpoints. In flat spacetime a straight inertial segment has the familiar endpoint interval, while accelerated paths between the same events can accumulate different proper times. Timelike geodesics locally maximize proper time within suitable neighborhoods, not necessarily globally after conjugate points or across arbitrary spacetime topology. Null curves accumulate zero proper time and require another parameter.

The bare coordinate measure changes by a Jacobian when coordinates change. The metric determinant supplies the inverse factor needed for a coordinate-invariant density. In four-dimensional Lorentzian signature the determinant is negative in a regular chart, so the density is the square root of minus that determinant. An oriented volume form additionally requires a choice of orientation; the positive density used for ordinary integration does not by itself choose one. A boundary integration also needs an induced measure and a consistent directed surface element.

Flat space in spherical coordinates illustrates the units. With x⁰ and r in metres and θ,φ dimensionless, the line element has angular coefficients r² and r²sin²θ. The four-volume density is r²sinθ on a chart with 0<θ<π and r>0. Its zeros at the polar-coordinate exclusions do not mean Minkowski geometry degenerates there. A regular Cartesian chart restores a nonsingular component matrix. In a purely spatial polar plane the corresponding area element is r dr dφ rather than dr dφ or r² dr dφ.

Light cones are local structures: they classify tangents and, with a time orientation, future causal directions. They do not by themselves determine whether a distant point is reachable or whether closed timelike curves exist. Those questions depend on the metric and topology over a region. Likewise the proper-time expression is invariant, but a coordinate time read by an arbitrary chart need not be a physical clock. Static or accelerated observer families require their own normalization by the metric.

The spacetime lab uses c=1, a fixed event position x=0.6, and a readout ct²−x², which is the opposite sign to our ds². Reset and set both controls to β=0 and ct=1, predicting transformed coordinates (x′,ct′)=(0.6,1). Reset and set β=0.6 and ct=1, predicting (0,0.8). In both runs its invariant readout is +0.64, corresponding to ds²=−0.64 in our convention. The lab is a single flat-space Lorentz transformation, not a curved metric or arbitrary worldline integrator.