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The Einstein Field Equations

Larry Smarr, NCSA/Univ. of Illinois, on-camera
Movie/Sound Byte
QuickTime Movie (2.3 MB); Sound File (1.6 MB); Text


Introduction

The Symbolic Form

Spherical Symmetry

Axisymmetry

Full Generality

Beyond the Math

Introduction

When expressed mathematically, Einstein's gravitation theory yields a set of formulas known collectively as the Einstein Field Equations. In a nutshell, this equation describes the properties of a gravitational field surroun ding a given mass. Stated simply, the equation describes how an object curves space and how the curvature, in turn, stretches or squeezes matter in three spatial directions: north-south, east-west and up-down.

However, applying the equations is no easy task. Mathematically, General Relativity is built upon ten so-called "coupled hyperbolic-elliptic nonlinear partial differential equations" which take many pages to write down (and a deep br eath just to say!). For a mathematician, the difficulties lie in the fact that the equations are nonlinear and coupled also and in their sheer number of terms.

After he published his famous paper in 1916, Einstein later conceded that the mathematical difficulties of his General Theory of Relativity were a "very serious" impediment to its further development. So serious, in fact, that it took nearly 75 years before the best minds in the field could come close to solving the equations stated by the theo ry. Now, high performance computers permit more accurate modeling of the distortions of spacetime by massive objects, including black holes.

Ed Seidel, NCSA/Univ. of Illinois, on-camera
Movie/Sound Byte
QuickTime Movie (1.2 MB); Sound File (788K); Text

Physicists are motivated to grapple with the complexity of Einstein's field equations because they can be used, theoretically at least, to describe all possible spacetime scenarios, from the collisions of black holes to the gravit ational interactions of irregular chunks of matter. In practice, researchers had to first restrict their calcuations to very simple systems in order to manage the equations.


The Symbolic Form

Field Equation

This elegant symbolic formulation of Einstein's general theory of relativity cannot be used for actual calculations, but it clearly shows the principle that "matter tells spacetime how to curve, and curved space tells matter how to move"(John Wheeler, Princeton University and the University of Texas at Austin) . The left side of the equation contains all the information about how space is curved, and the right side contains all the information about the location and motion of the matter. General relativity is beautiful and simple (to a physicist), but mathemat ically it's very complicated and subtle.

Larry Smarr, NCSA/Univ. of Illinois, on-camera
Movie/Sound Byte
QuickTime Movie (1.8 MB); Sound File (961K); Text


Spherical Symmetry

The first person to apply Einstein's general theory was the German astrophysicist Karl Schwarzschild. It was 1916, and the First World War was still raging. Just a few days after reading Einstein's newly published theory while stationed at the Russian front, Schwarzschild began to figure out its consequences for the gravitational fields of stars.

For his first attempt, Schwarzschild simplified the problem by considering a perfectly spherical star at rest, ignoring the effects of the star's interior. He sent his preliminary solution of the star's spacetime curvature to Einstein, who reported the results on Schwarzschild's behalf at a physics meeting in January. The curva ture of spacetime predicted by the solution became known as the Schwarzschild geometry and had profound implications on future research into gravitation and cosmology.

A few weeks later, Schwarzschild sent a second paper, this time describing the spacetime curvature inside a star. Tragically, Schwarzschild died a few months later of an illness he contracted while at the Russian front.

Schwarzschild was describing a singularity, a region of infinite spacetime curvature that is postulated to lie within what has more recently been termed a black hole. Einstein considered the "Schwarzschild singularity" and black holes as biz zarre constructs, resisting the logic of his own theory right up to his death in 1955.

However, though debate continues on the nature of singularities, since the 1960s there has been mounting indirect evidence that black holes might exist in places where, for example, a collapsed star's intense gravitational field allows nothing, including light, to escape. In that sense, the star disappears from the visible universe and forms what is now called a black hole.

Had Schwarzschild lived, he would have likely developed more elaborate scenarios of spacetime curvature. Nevertheless, his early achievement was not topped for nearly 50 years, when the simple, spherical geometry of his solution was finally expanded to co nsider the gravitational effects of a spinning star.

Now, in the 1990s, astrophysicists are using supercomputers to extend the calculations to more complex spacetime geometries, including spinning objects that no longer retain their simple, spherical symmetry. Such objects typically exhibit "axisymmetry" -- symmetry about one axis (like a football) -- or no symmetry at all. The mathematics, however, becomes very difficult to state, let alone solve analytically.


Axisymmetry

If one considers an axisymmetric object, one that exhibits symmetry about a single axis, the equations expand dramatically into dozens of pages.

Einstein Equations in 2D
Long form

At this point, the equations are so complex they can only be manipulated reliably by special computer software. Advanced computational techniques are needed to study distorted, spinning black holes, or the head-on collision of two axisymmetric black holes. This exhibit presents the results of many such black hole calculations.


Full Generality

If we consider yet less idealized systems, without any symmetries to reduce the problem's complexity, the problem explodes into equations with literally thousands of terms in each equation. The following example merely shows a single expression from just one of the equations written in Fortran computer code.

Einstein Equations in 3D
Fortran code

JPEG (132 KB)

In three-dimensional, time-dependent spacetimes, the lack of symmetry results in yet more variables. Just for the gravitational field itself, there are at least 16 variables in the standard formulation (developed for computation by Arrow itt, Deser and Misner in 1962). These variables depend on further coordinates, three for space and one for time, so there are many, many more terms in the equations.


Beyond the Math

Fortunately, it is possible to apply the General Theory of Relativity to the "real world" without having to completely solve the mathematics. For one thing, we can try to find examples in the universe where objects or processes exhibit the behaviors that the theory predicts. In fact, several observational tests of the General Theory of Relativity were proposed by Einstein and others almost immediately after its publication in 1916.

Forward to Putting Relativity to the Test
Return to General Relativity
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NCSA. Last modified 11/7/95.