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. 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.
thumb|Figure 1. Tidal effects.
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. 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.
thumb|Figure 2. Equivalence principle
- 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 (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in.
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.
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 <math>ds^2</math> hence depends on both the intrinsic properties of the surface and the coordinate system used to describe that surface. A cursory examination of <math>ds^2</math> 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.
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.
Years before publication of the general theory in 1916, Einstein used the equivalence principle to predict the existence of gravitational redshift in the following thought experiment: (i) Assume that a tower of height h (Fig. 4) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity , so that its total energy E, as measured by an observer on the ground, is (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E back into a particle of rest mass m.
Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock. For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.
Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established the distortion of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the space component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.
Newtonian gravitation can thus be described informally as a "curvature of time", although strictly speaking, "curvature" is a property of spacetime as a whole rather than of time alone. General relativity, on the other hand, is a theory of curved spacetime. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable: The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.
thumb|Figure 5. General relativity is a theory of curved time and curved space. [[:File:General relativity time and space distortion extract.gif|Click here to animate. ]]
As the astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing irregularities in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.
In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.
The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.
Sources of spacetime curvature
thumb|250px|Figure 6. Contravariant components of the stress–energy tensor
In Newton's theory of gravitation, the only source of gravitational force is mass.
In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations,
<!-- : <math>R_{\mu \nu} - \tfrac{1}{2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}</math> -->
the sources of gravity are presented on the right-hand side in <math>T_{\mu \nu},</math> the stress–energy tensor.
One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity.