Spacetime: Space and Time United

The fusion of space and time dimensions

Before the 20th century, space and time were considered distinct entities: space was absolute, structuring distances, while time flowed uniformly. This Newtonian view was overturned by Einstein's theory of special relativity (1905), which demonstrated that measurements of space and time depend on the observer's frame of reference. Henceforth, events must be described in a unified four-dimensional framework: three of space and one of time.

Space is time; time is space

An inseparable geometric relationship

In special relativity, space and time are intimately linked dimensions. Their relationship is expressed by the invariance of the spacetime interval \(\Delta s^2\), which remains constant for all inertial observers: \[ \Delta s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 \]

This means that an increase in the spatial component results in a decrease in the temporal component, and vice versa. This compensation is at the heart of the principle of relativistic causality: two events can only be causally linked if the spacetime interval separating them is of the "temporal" type, i.e., if \(\Delta s^2 > 0\). In this case, a signal or information, even traveling at the speed of light, can connect the two events without violating the limit imposed by relativity. On the other hand, if the interval is of the "spatial" type (\(\Delta s^2 < 0\)), no causal link is possible: the temporal order of events can even differ according to observers, making their relationship non-physical. This distinction delineates the light cone, the boundary between what can influence or be influenced, and what is definitively out of causal reach.

The light cone and the tilt of the present

An often-used analogy is that of the light cone: for a moving observer, this cone is tilted, modifying the distribution between the space and time axes. In this sense, what one considers a spatial interval, another perceives as a temporal interval. This is particularly evident in the phenomenon of time dilation: a moving observer will measure a longer duration for the same phenomenon, as time has contracted in their frame of reference, while distance has extended.

Extreme situations: black holes and inflation

Even more astonishingly, in black holes or in the expanding universe, this interchangeability becomes extreme: at the horizon of a black hole, time "freezes" for the distant observer, while the radial coordinate becomes temporal. In the primordial universe, where the expansion of space is rapid, cosmic time flows more slowly as space "grows." It is a true compensated dynamic: spatial expansion absorbs time.

A compensated exchange between two conjugate dimensions

Thus, in relativistic physics, space is not independent of time: they are two faces of the same entity. A variation in one implies a response in the other, somewhat like two conjugate variables of a constant-sum system. One might say: "when space extends, time slows down."

The spacetime continuum

Spacetime is a geometric entity where each point is an event, defined by its coordinates \((x, y, z, t)\). This mathematical framework allows us to understand relativistic physical phenomena. For example, simultaneity becomes relative: two events judged simultaneous in one frame of reference may no longer be so in another.

General relativity and the curvature of spacetime

In 1915, Einstein extended his theory by integrating gravitation. He postulated that masses deform the fabric of spacetime, an idea expressed in Einstein's equation: \[ R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \] This equation links geometric curvature (left) to the distribution of energy and matter (right). Thus, gravity is no longer a mysterious force at a distance, but a geometric manifestation.

Measurable consequences

The concept of spacetime allows us to predict measurable phenomena: deflection of light by stars (lensing), time dilation (slower time in strong gravity), and even the existence of gravitational waves, first detected in 2015 by LIGO. These ripples in spacetime confirm that it is dynamic, deformable, undulating like a cosmic membrane.

Spacetime: A complex topological structure

On a quantum or cosmological scale, spacetime could present even more exotic structures: wormholes, quantum fluctuations, or "foam" of spacetime according to quantum gravity. These researches are at the frontier of theoretical physics, between general relativity and quantum mechanics.

Sources: Einstein Papers Project, LIGO Caltech, Scientific American – Einstein & Spacetime.