1The theory of general relativity is the theory of the disappearance of time. What does this disappearance mean? And what notions of a temporal nature does the theory allow us to consider?
2I begin with a historical analogy. If, as we know, for Aristotle there is no physical space as such, we can characterize the geometric framework of his physics by the concepts of horizontal and vertical. The two are different in nature as, for example, the natural movement of heavy bodies follows the vertical (in the sub-lunar world). This corresponds to the common sense observation that it is easier for an object to move horizontally than vertically, which today we attribute to gravity. The Newtonian revolution of the seventeenth century then reads as the following substitution:
4Indeed, one of Newton’s major contributions was the introduction of a homogeneous and isotropic physical space, represented mathematically by Euclidean geometric space. Isotropy means that all dimensions are the same: in particular, nothing distinguishes vertical from horizontal. This was made possible only by recognizing the phenomenon responsible for the apparent distinction between these dimensions: gravitation. And it is well known that introducing this interaction was another of Newton’s great innovations. The scientist’s genius was to have declared, contrary to appearances, that the horizontal and vertical dimensions were of the same nature. It was against all appearances because in everyday life (on Earth), everything seems to distinguish them from one another, and Newton was unable to benefit from the experiments in weightlessness that make this disappearance clear today. What is vertical for an astronaut in a spaceship? Where would it be for an inhabitant of a planet in the Andromeda Galaxy? This could, of course, be defined within its immediate vicinity, for example, directed towards the center of its planet, but this would have nothing to do with our “vertical” on Earth, which I would describe as our “proper vertical” (which is also ambiguous: vertical in relation to the pole or to the equator?). In Newtonian space, there is no longer only one vertical, but each can define his or her own proper vertical, for example, simply as the direction running from one’s head to one’s toes.
5The relativist revolution, that of special relativity (SR), is formulated in a completely identical manner, in which “vertical” is replaced with “time” and “horizontal” with “space.”
7Spacetime is isotropic: all directions are identical. None of them represents time; none represents space (directions split however in three families: timelike, spacelike or lightlike). It is only in our local environment that we can distinguish a particular dimension we call time. It is proper to us, and defined only at the position that we occupy in spacetime. It is this dimension that connects our present state to our future state. But it only applies to us, and it only has relevance within our environment (strictly speaking, only for the position we occupy; with limited accuracy, for the extended neighborhood of the Earth, the solar system, or even our galaxy).
8Another “observer” experiences the flow of his or her own proper time. The latter has no more to do with ours that the vertical of an inhabitant of Andromeda has to do with the terrestrial vertical:  there is no one time valid for everyone and everywhere, any more than there is one single vertical in the totality of space.
9This is inscribed deep within the mathematical formulation of the theory. The relevance of the geometric framework of physics (in other words, of nature) is demonstrated by the transformations that we can make within it. These transformations form a group in the mathematical sense of the term.
10– With Aristotle, horizontal rotations (that is to say, within a plane) form a group that mathematicians call SO(2), in which the 2 refers to the two dimensions of the plane in which it operates. A rotation within this plane is defined solely by its angle: this group is considered to be one-dimensional. Rotations within the vertical dimension are, for their part, trivially limited.
11– With Newton, the group widens: all spatial rotations are permitted and equivalent, including those involving the vertical. And they all have the same status. We can observe this in concrete terms nowadays in spaceships (think of Captain Haddock’s drop of whiskey [in Hergé’s Explorers on the Moon]: in zero-gravity conditions, it assumes a spherical shape that directly shows the isotropy of space). The group of rotations in space is larger: it has three dimensions. Mathematicians call it SO(3), in which the 3 refers to the three dimensions of space.
12The transition to Newtonian physics is therefore expressed by an increase in the symmetries of the geometric framework. A posteriori, the vertical appears as a “breaking” of the symmetry of space, reducing the group SO(3) to SO(2) only.
13– With Einstein (SR), the symmetry is even greater since all dimensions have the same status, including those involving what we would still call time. The group is now the group of rotations in spacetime. It is called SO(3,1) or the Lorentz group. It is six-dimensional. The existence of this group directly indicates that it is impossible to extract just one of the dimensions of spacetime and call it time; all of them have the same status (at least within this big family). The symmetries of spacetime are much more numerous than those of space.
14The isotropy of spacetime—the impossibility of selecting out a temporal dimension naturally—is the basis of SR, just as the isotropy of space—the impossibility of defining just one vertical—is the basis of Newtonian physics. In retrospect, the Newtonian division of space from time also appears as a “breaking of symmetry”: time (for Newton) is what breaks the symmetry of spacetime (in SR) just as the vertical is what breaks the symmetry of space.
15General relativity goes even further. It is not just all rectilinear dimensions of spacetime that have the same status: all curves (at least those of the kind specifically called “timelike”) are equivalent. None of these curves (which would one choose?) can be called time.
16In historical terms, this spatio-temporal vision was not immediately accepted. What Einstein was the first to see was the disappearance of time. It was only later, following the work of Minkowski and Poincaré, that the concept of spacetime was introduced. Einstein deduced the disappearance of time from the disappearance of simultaneity. Via thought experiments involving light signals, he realized that it was impossible to decide whether two events were simultaneous or not. If a single time existed, this would allow us to date events. And if we could date events, it would be sufficient to compare dates to decide whether or not they are simultaneous.
17It should be noted that Einstein developed a thought experiment that allowed a given observer to determine the simultaneity of two events from his or her personal point of view. This relative simultaneity is only definable and meaningful as a result of some measurement made by the observer that we take account of (and not another). It is not thus far associated with the proper time of this observer (nor to any other). Einstein clearly demonstrated that two events that appear simultaneous to an observer will not appear simultaneous to one another. To explain it more clearly (if clarification were needed): it is impossible to define just one time (or even a temporal function) that would be such that these simultaneities would correspond to similar dates.
Causality and Proper Times
18If there is no time in relativity (special or general), the theory does admit two fundamental concepts that have a temporal connotation: causality and proper times. In Newtonian physics, these two notions merge with that of time: causality is identical to the chronology established via time, and all the proper times between two events are identical, together with the “lapse of time” marking the difference between the moments of the two events. The confusion between these relativistic concepts and the concept of time is a major source of misunderstanding of relativity.
19To understand the concept of proper times, consider two events in the universe: A, the explosion of a distant star, and B, the arrival of a meteorite on Earth. In Newtonian physics, time is used to assign a date to each. The duration (time) between these two events is then defined simply as the difference in their dates. For example, explosion A took place 1000 years ago, the arrival of the meteorite 100 years ago; the duration that separates them is 900 years. All this has a definite meaning in Newtonian physics. In relativity, there is no notion of a duration between these two events. The theory considers them as two points A and B in spacetime between which we can plot (or imagine plotting) with an infinite number of curve segments connecting them. Specifically, the metric measure of spacetime (actually, a pseudo-metric measure: one of its fundamental properties in SR) allows us to assign a (pseudo-)length to each segment of the curve. This is what the theory calls the proper time separating these two events along the corresponding curve. Note that this is the only concept close to that of (usual) duration that can be defined within the framework of the theory. We do not define the “proper time between A and B” but “proper times between A and B along each segment of the curve joining them.” There are as many of these as there are of such segments: infinite numbers of them.
20Some of these segments may be associated with observers. An observer occupies, at each instant of his or her life, a point in spacetime. By joining all the points in his or her history, we get a line in spacetime: the world-line, a continuous succession of the events of his or her life. Let us imagine that an observer has experienced both of the events of which we have just been speaking (in relativity, this is a necessary condition if we are to be able to define a proper time between them). This means that the world-line passes through points A and B. The proper time thus associated with the observer’s history is that which he or she has experienced, felt, and measured between the two events.
21There are many possible observers, many world-lines, many proper times between two specific events (provided that they have actually participated in them; otherwise, they cannot define any proper time). Two observers with different histories will have experienced two different proper times between A and B. None of them defines something that we could call “time” that took place between the two events. 
22Langevin’s anecdote of the twins illustrates the nature of proper time. At the age of 18, the twins separate: Isaac stays on Earth while Albert goes on an interstellar trip. When Albert (the traveling twin) returns, he celebrates his reunion with his brother. Isaac is 50 years old, while Albert is only 30. The twins have had two different histories; they have experienced, felt, and measured different proper times. It should be noted that there is no ambiguity: in every sense of the word, Albert is 30 years old, Isaac is 50. Albert has really lived for 32 years, Isaac 12 years: whether they measure their (proper) years by their watches, by the beating of their hearts, or by using a stopwatch to measure the (proper) time they require to read a book or to do a mental calculation….
23Isaac, the sedentary twin has not moved (or very little): his world-line is a straight line (SR) or a geodesic (in GR). This is not the case for Albert who, in order to depart and then to turn around, has experienced the thrust of the engines of the rocket: he has followed a complex curve in spacetime. Whether the problem is analyzed from the perspective of SR or GR, the result is the same: the twins have had two different experiences of time.
Universal Time and Cosmic Time?
24We still have to understand how it can be that we have the feeling of a universal time that would elapse for everyone in Newtonian fashion. The theory explains this perfectly. If the curvature of spacetime is not too great (in other words, if the gravitational field is not too strong), and if one is only interested in movements that are not too fast (compared to the speed of light), then the validity of an observer’s proper time can be extended over a small surrounding area.
25To what extent? Everything depends on the accuracy of the measurements under consideration (in relation to the intensity of the gravity and speeds involved). For all the purposes of everyday life, I can extend the validity of my proper time to the scale of the Earth, and even the solar system, without committing any detectable error. The GPS tracking system already requires greater accuracy. Its data cannot be converted into reliable information without taking into account the fact that one cannot speak of time at a scale shared by the Earth and the satellites orbiting it; the Earth has its proper time, and the satellites have theirs. The system for analyzing their data takes account of this difference. If more precision is required, it is impossible to assume that a single proper time (even relative) extends to the entire Earth. The latest versions of atomic clocks, placed a few meters apart from one another, do not measure a single time that flows at the same rate for both; rather, each measures a different quantity from the other, the flow of its own proper time. They cannot be synchronized.
26Another example of a temporal nature is that of cosmic time in cosmology. The spacetimes that constitute cosmological models are assumed to be relatively simple (respecting the cosmological principle). This simplicity—in fact, the existence of important symmetries—allows us to define a temporal function for all of spacetime, called cosmic time. Some of its properties are reminiscent of time—hence the name—and for ourselves (who have defined it), it identifies (approximately) with our proper time along our world-line. It is a useful tool for analyzing cosmic processes, but we should remain cautious because, for example, two events occurring at the same value of cosmic time are not seen as simultaneous, even by ourselves (who are nonetheless the best-positioned observers). And indeed, in general, we have no idea of the value of the cosmic time of an event that we observe: a particular value of cosmic time associated with an event is the result of an indirect and often imprecise restoration, which is generally not mentioned. Indeed, the real utility of this “cosmic time”—without much epistemological relevance—is to let us reconcile a popularized cosmological discourse with familiar concepts (often, unfortunately, at the price of imprecision).
27Finally, I will conclude with the notion of causality, which is very important in physics. In Newtonian physics, causality is “trivial” in the sense that it fuses with chronology. In relativity, there is no chronology but a perfectly defined causality, so much so that many physicists consider it to be the fundamental property of spacetime. To explain, let us consider two events, A and B: A may causally precede B, and B may causally precede A, or A and B may be causally disconnected. In the latter case, the two disconnected events A and B will be declared simultaneous by one observer; but A may precede B from the chronological point of view of a second observer, and B may precede A from the chronological point of view of a third. In other words, causality and temporality are two entirely different concepts in Einsteinian physics.
28The concept of time is fundamentally incompatible with relativity. And the theory is better formulated if we avoid references to any temporal notion. Some new approaches in fundamental physics, and especially quantum gravity theories, go further in “cleaning up” physics (in particular quantum physics) by formulating it without reference to time. But that is a tale for another time.
Even if it occupies the same position as us; in this case, however, if it moves in relation to us.
There is a maximum value to all of these times, which corresponds to a particular curve (a history) joining the two events: a geodesic. An observer can describe a geodesic provided that its movement is inertial, in other words, that it is not subjected to any force: no engine, no contact with the ground, etc.