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Where v is velocity, x, y, and z are Cartesian coordinates in 3-dimensional space, c is the constant representing the universal speed limit, and t is time, the four-dimensional vector v = (ct, x, y, z) = (ct, r) is classified according to the sign of c 2 t 2 − r 2. A vector is timelike if c 2 t 2 > r 2, spacelike if c 2 t 2 < r 2, and null or ...
A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2-1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light ...
Spacetime algebra is a type of geometric algebra that is closely related to Minkowski space, and is equivalent to other formalisms of special relativity. It uses mathematical objects such as bivectors to replace tensors in traditional formalisms of Minkowski spacetime, leading to much simpler equations than in matrix mechanics or vector calculus.
Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy , as expressed in the mass–energy equivalence formula E = m c 2 {\displaystyle E=mc^{2}} , where c {\displaystyle ...
where X = Y = Z = 0. Alternatively, since we wish to build a Lagrangian out of relativistically invariant quantities, take the action as proportional to the integral of the Lorentz invariant line element in spacetime, the length of the particle's world line between proper times τ 1 and τ 2, [nb 1]
In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study.The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.
After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications of general relativity. A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3).
In this world, the difference in height between two objects at the same x, y would be a rotationally invariant quantity unrelated to length and width. The z-coordinate is effectively separate from x and y. Eventually, experiments at large angles would convince the creatures that the symmetry of the world is SO(3). Then they would understand ...