frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions. Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing (representing an observer) and another containing (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally. For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented. Geodesics are said to be timelike, null, or spacelike if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by timelike and null (lightlike) geodesics, respectively. Privileged character of 3+1 spacetime See also Basic introduction to the mathematics of curved spacetime Complex spacetime Einstein's thought experiments Four-dimensionalism Global spacetime structure Metric space Philosophy of space and time Present Notes Additional details References Further reading George F. Ellis and Ruth M. Williams (1992) Flat and curved space–times. Oxford Univ. Press. Lorentz, H. A., Einstein, Albert, Minkowski, Hermann, and Weyl, Hermann (1952) The Principle of Relativity: A Collection of Original Memoirs. Dover. Lucas, John Randolph (1973) A Treatise on Time and Space. London: Methuen. Chpts. 17–18. External links Albert Einstein on space–time 13th edition Encyclopædia Britannica Historical: Albert Einstein's 1926 article Encyclopedia of Space–time and gravitation Scholarpedia Expert articles Stanford Encyclopedia of