Spacetime

Non-relativistic classical mechanics treats time as a universal quantity of measurement that is uniform throughout, is separate from space, and is agreed on by all observers.

A point in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1).

Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location.

[8] George Francis FitzGerald in 1889,[14] and Hendrik Lorentz in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment.

[16]: 163–174  Lorentz's equations predicted a quantity that he called local time, with which he could explain the aberration of light, the Fizeau experiment and other phenomena.

[21][22] He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world".

[16]: 163–174 In 1905, Albert Einstein analyzed special relativity in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics.

His work was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.

By using the mass–energy equivalence, Einstein showed that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing general relativity.

He never made a priority claim and always gave Einstein his full share in the great discovery.Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. Minkowski saw Einstein's work as an extension of Lorentz's, and was most directly influenced by Poincaré.

[26] On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title, The Relativity Principle (Das Relativitätsprinzip).

[note 5] On 21 September 1908, Minkowski presented his talk, Space and Time (Raum und Zeit),[27] to the German Society of Scientists and Physicians.

1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.

[note 7] Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness).

Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line.

Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:[38] A pure SR analysis would be as follows: Analyzed in Stella's rest frame, she is motionless for the entire trip.

The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.

To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light, Fig.

The invariant hyperbola has the equation x = √w2 + k2, where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v.

The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls.

Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously.

3-5 is based on one presented by Bais[41]: 64–66  and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections.

To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.

Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time.

It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.

In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity.

[59]: 397–403 The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate.

In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner[60] and Rainer K. Sachs[61] addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves.

The implicit notion that the dimensionality of the universe is special is first attributed to Gottfried Wilhelm Leibniz, who in the Discourse on Metaphysics suggested that the world is "the one which is at the same time the simplest in hypothesis and the richest in phenomena".

On the other hand, in view of creating black holes from an ideal monatomic gas under its self-gravity, Wei-Xiang Feng showed that (3 + 1)-dimensional spacetime is the marginal dimensionality.

Figure 1-1. Each location in spacetime is marked by four numbers defined by a frame of reference : the position in space, and the time, which can be visualized as the reading of a clock located at each position in space. The 'observer' synchronizes the clocks according to their own reference frame.
Figure 1–4. Hand-colored transparency presented by Minkowski in his 1908 Raum und Zeit lecture
Figure 2–1. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C
Figure 2-2. Galilean diagram of two frames of reference in standard configuration
Figure 2–3. (a) Galilean diagram of two frames of reference in standard configuration, (b) spacetime diagram of two frames of reference, (c) spacetime diagram showing the path of a reflected light pulse
Figure 2–4. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere"
Figure 2–5. Light cone in 2D space plus a time dimension
Figure 2–6. Animation illustrating relativity of simultaneity
Figure 2–7. (a) Families of invariant hyperbolae, (b) Hyperboloids of two sheets and one sheet
Figure 2–8. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds
Figure 2–9. In this spacetime diagram, the 1 m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.
Figure 2–11. Spacetime explanation of the twin paradox
Figure 3–1. Galilean Spacetime and composition of velocities
Figure 3–2. Relativistic composition of velocities
Figure 3-3. Spacetime diagrams illustrating time dilation and length contraction
Figure 3–4. Lorentz factor as a function of velocity
Figure 3–5. Derivation of Lorentz Transformation
Figure 3–6. Spacetime diagram of relativistic Doppler effect
Figure 3–7. Transverse Doppler effect scenarios
Figure 3–8. Relativistic spacetime momentum vector. The coordinate axes of the rest frame are: momentum, p, and mass * c. For comparison, we have overlaid a spacetime coordinate system with axes: position, and time * c.
Figure 3–9. Energy and momentum of light in different inertial frames
Figure 3–10. Relativistic conservation of momentum
Figure 5–1. Tidal effects.
Figure 5–2. Equivalence principle
Properties of ( n + m ) -dimensional spacetimes [ 65 ]