Allgemeine Relativitätstheorie

Die allgemeine Relativitätstheorie im Cube of Physics.

In order to unite Newtonian gravity with special relativity, both natural constants \(G\) and \(c^{-1}\) must be taken into account. This is exactly what Einstein’s General Theory of Relativity (1915) does. It transforms the static four-dimensional Minkowskian space-time into a dynamic and curved object, whose shape is uniquely fixed by Einstein’s field equations. In curved space-time, objects always move on the shortest possible path as determined by an indefinite metric. A historically important example is Mercury’s orbit around the Sun.

According to Einstein, the Sun curves space-time in a way that results in a nearly elliptic orbit, up to a small perihelion shift. Contrary to Newton, the reason for Mercury’s curved path is not some mysterious gravitational force emanating from the Sun, but simple economy. An analogy is the trajectory of an airplane from Berlin to New York: it is curved, but by no means due to a hypothetical force at the North Pole.

Why was Albert Einstein so sure that he had to develop a new theory of gravity? In fact, Max Planck was not aware of the cube’s missing fourth classical corner, corresponding to \(c^{-1}\) and \(G\). Accordingly, Planck \emph{strongly recommended} him to not waste his time, but to rather focus on the development of quantum mechanics. However, Einstein was acutely aware that Newtonian gravitational theory and special relativity are fundamentally incompatible. For example, in the former, a change in the distance between two masses results in an instantaneously felt change in the gravitational force acting between them. In contradistinction, in special relativity signal transmission at velocities greater than c is impossible. This fundamental feature, that Newtonian gravity depends merely on location but not on time, had to be eliminated by general relativity.

In hindsight, the most elegant way to derive the latter theory involves a suitable action principle. Let us first recall the much simpler derivation of Newton’s laws based on such a principle. One considers a body’s movement over a period of time and to determine and determines the actual path traveled according to a suitable criterion. »The path of actual reproduction is that to which the smallest amount of action belongs!« (Maupertuis 1744) This extreme value principle then implies the famous Euler-Lagrange equations:

\[\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}-\frac{\partial L}{\partial q_i}=0\]

Here, \(q_{i}(t)\) is the coordinate of the i-th particle at time \(t\), and \(L\) the Lagrangian, given by the difference between the particles’ kinetic and potential energies. These precisely reproduce Newton’s equations of motion. Incidentally, note that this derivation is based on an erroneous fundamental asymmetry of space and time: the different states of a particle are integrated over time.

Generalising this idea to Einsteinian gravity, one is looking for a suitable action \(S\) that allows for its minimisation under arbitrary variations of the metric:

\[\frac{\delta S}{\delta g_{\mu\nu}}=0\]

The latter is then once again calculated by using an analogue of the Euler-Lagrange equations. The correct choice is now called the Einstein-Hilbert action and leads to Einstein’s field equations:

\[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi G}{c^4}T_{\mu\nu}\]

They describe in a mathematically rigorous fashion that matter (\(T_{μν}\)) bends (\(R_{μν}\) and \(R\)) space-time (\(g_{μν})\). Particles move in this metric on curved tracks without experiencing any force: gravity simply corresponds to the curvature of space-time.

Therefore, in the following we want to assume the complete physical equivalence of gravitational field and corresponding acceleration of the reference system.

The objects that move in the cosmos thus follow only one goal in their orbits: the shortest and fastest connection in a curved space-time. On their way they bend with their mass and energy the space-time in their immediate vicinity. Everything is in motion, everything influences this dynamic and highly non-linear manifold. General relativity solves the contradictions between Newton’s theory of gravity and special relativity by making both constants \(G\) and \(c^{-1}\) relevant at the same time: as the theory (\(G\), \(c^{-1}\), 0) of the cube.

In 1916, Einstein also predicted in his General Theory of Relativity that gravitational waves must exist in analogy to electromagnetic waves. But their experimental proof seemed unrealistic to him. In 2016, this proof was achieved for the first time with the Laser Interferometer Gravitational Wave Observatory (LIGO). The origin of gravitational waves could be explained by the collapse of a binary system of black holes.

LIGO measurement of gravitational waves at Hanford and Livingston compared with the theoretically predicted values.