In physics, curved spacetime is the mathematical model in which, with Einstein's theory of general relativity, gravity naturally arises, as opposed to being described as a fundamental force in Newton's static Euclidean reference frame. Objects move along geodesics—curved paths determined by the local geometry of spacetime—rather than being influenced directly by distant bodies. This framework led to two fundamental principles: coordinate independence, which asserts that the laws of physics are the same regardless of the coordinate system used, and the equivalence principle, which states that the effects of gravity are indistinguishable from those of acceleration in sufficiently small regions of space. These principles laid the groundwork for a deeper understanding of gravity through the geometry of spacetime, as formalized in Einstein's field equations.
Introduction 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. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it." In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.
In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.
In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. 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.
Different observers viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first observer, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second observer, aware of the large mass 1, smiles at the first reporter's naiveté. This second observer knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third observer, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.
Two central propositions underlie general relativity. The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion." This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.
The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration. In Fig. 2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable, there are no experiments that A and B can perform which will enable them to tell which setting they are in. An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, mgg and in Newton's second law, there is no a priori reason why the gravitational mass mg should be equal to the inertial mass mi. The equivalence principle states that these two masses are identical.
To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations.
What curvature means The spacetime interval s2 The "spacetime interval" (or "invariant interval") is used as an invariant "distance" between two events as measured by different observers.
It acquires additional importance in general relativity, since the form of this interval serves to characterize the geometric properties of the space (or spacetime) on which the interval has been defined. For example, in a Euclidean plane, the Pythagorean theorem holds for right triangles drawn in that plane. Conversely, if the distance between two points on a surface is given by the Pythagorean theorem, then that surface is necessarily a Euclidean plane.
Intrinsic curvature
In curved surfaces, the Pythagorean theorem generally does not hold in its ordinary Euclidean form. Intrinsic curvature can be determined entirely by measurements made within the surface itself, without reference to any higher-dimensional embedding.
Intrinsic curvature does not require a surface to be embedded in a higher-dimensional space. If a flat sheet is rolled into a cylinder or folded into complex origami forms without stretching, such deformations alter the sheet’s extrinsic curvature while leaving its intrinsic geometry unchanged.
In general relativity, spacetime curvature is treated intrinsically through measurements made entirely within spacetime itself of such effects as geodesic deviation and tidal effects (described above in the Introduction).
Curvature versus coordinates
In relativistic calculations, one often chooses coordinate systems that simplify the computations. The expression for the Pythagorean theorem, and hence the spacetime interval, depends both on the geometry of spacetime and on the coordinate system used to describe it. For example, in Fig. 3, we see for the following coordinate systems:
In cases (a), (b), and (c), the surfaces are flat. In case (d), the surface is spherical.
The expression for hence depends on both the intrinsic properties of the surface and the coordinate system used to describe that surface. A cursory examination of will not suffice to determine the characteristics of the surface that we are dealing with, since the appearance of the metric coefficients alone does not determine whether a space is intrinsically curved. Complicated coefficients may arise merely from the choice of coordinates. Additional mathematical quantities are needed to distinguish genuine curvature from coordinate artifacts.
One such quantity is the Christoffel symbol, which describes how coordinate grids change from point to point. Christoffel symbols may become nonzero even in flat space when curvilinear coordinates are used. They therefore do not by themselves represent intrinsic curvature.
Intrinsic curvature is instead characterized by the Riemann curvature tensor. In flat spaces, the components of the curvature tensor vanish even when the metric coefficients and Christoffel symbols are complicated. Nonzero curvature tensor components indicate genuine curvature that cannot be eliminated by a change of coordinates.
Local flatness and inertial frames
Spacetime in general relativity is generally curved because of gravity. Nevertheless, every sufficiently small region of spacetime approximates flat Minkowski spacetime. Within such a small region, the laws of physics reduce to those of special relativity.
Mathematical quantities called Christoffel symbols describe how coordinate systems change from point to point. These quantities may be interpreted as representing local gravitational and inertial effects. Einstein's thought experiment of the falling painter illustrates that one can always choose a freely falling frame in which such local effects disappear. Mathematically, one can always choose coordinates in a sufficiently small region of spacetime such that the Christoffel symbols vanish at a point.
Spacetime curvature, however, represents an intrinsic geometric property of spacetime associated with tidal effects and geodesic deviation. Unlike Christoffel symbols, curvature cannot generally be transformed away even locally by a change of coordinates. Even in a freely falling frame where local gravitational effects disappear, nearby freely falling particles may still accelerate relative to one another because of spacetime curvature.
Thus, the equivalence principle states that gravity can always be locally transformed away in a sufficiently small freely falling frame, while spacetime curvature remains as the underlying geometric property of spacetime.
Newtonian gravity and gravitational time dilation
In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.