Dark energy in its simplest formulation takes the form of the cosmological constant term in Einstein's field equations of general relativity, but its composition and mechanism are unknown and, more generally, the details of its equation of state and relationship with the Standard Model of particle physics continue to be investigated both observationally and theoretically.

Ten years later, Alexander Friedmann, a Russian cosmologist and mathematician, derived the Friedmann equations from Albert Einstein's equations of general relativity, showing that the Universe might be expanding in contrast to the static Universe model advocated by Einstein at that time.

Solutions of Einstein's equations that violate this inequality exist, but they do not possess an event horizon.

Singularities that arise in the solutions of Einstein's equations are typically hidden within event horizons, and therefore cannot be seen from the rest of spacetime.

These equations specify how the geometry of space and time is influenced by whatever matter is present, and form the core of Einstein's general theory of relativity.

With this additional condition — the covariant divergence of the energy – momentum tensor, and hence of whatever is on the other side of the equation, is zero — the simplest set of equations are what are called Einstein's ( field ) equations:

At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, pseudo-Riemannian manifold representing spacetime, and the energy – momentum contained in that spacetime.

The core concept of general-relativistic model-building is that of a solution of Einstein's equations.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy – momentum tensor must be divergence-free.

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.

Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult.

It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity ( encoded in the nonlinearity of Einstein's equations ).

Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.

with R < sub > AB </ sub > the five-dimensional Ricci curvature, may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations

Similarly, the set of all solutions to Einstein's field equations is simpler than a specific solution.

Even after experiments showed that Einstein's equations for the photoelectric effect were accurate, resistance to the idea of photons continued, since it appeared to contradict Maxwell's equations, which were well-understood and verified.

Again, the weighing of evidence and importance of new data was fit through the human sieve: some scientists found the simplicity of Einstein's equations to be most compelling, while some found them more complicated than the notion of Maxwell's aether which they banished.

There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.

* the equivalence principle, whether or not Einstein's general theory of relativity is the correct theory of gravitation, and if the fundamental laws of physics are the same everywhere in the universe.

If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox ( although Bell's theorem would still be valid ).

Therefore, as historians such as John Stachel argue, Einstein's views on the " new aether " are not in conflict with his abandonment of the aether in 1905.

As a consequence of Einstein's theory of special relativity, electricity and magnetism are fundamentally interlinked.

Crew members are spelling out Albert Einstein | Einstein's mass-energy equivalence formula E = mc < sup > 2 </ sup > on the flight deck.

Pantheism ( All-is-God ) is often associated with monism ( All-is-One ) and some have suggested that it logically implies determinism: in Einstein's words, " the past, present, and future are an ' illusion '".

Not to be confused with gravity waves, gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

Scientists also use thought experiments when particular physical experiments are impossible to conduct ( Carl Gustav Hempel labeled these sorts of experiment " theoretical experiments-in-imagination "), such as Einstein's thought experiment of chasing a light beam, leading to Special Relativity.

" The fundamental idea of 1924 thesis was the following: The fact that, following Einstein's introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated ... My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons.

Except for the Dym equation and the Ginzburg – Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier – Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.

They are particularly important in relativity theory, where they both appear on the side of Einstein's field equations that represents the geometry of spacetime ( the other side of which represents the presence of matter and energy ).

There is no need to invoke Einstein's theory of special relativity, because all observations are made in the same frame of reference.

Our modern conception of time is based on Einstein's theory of relativity and Hermann Minkowski's spacetime, in which rates of time at separate places run differently, and space and time are merged into spacetime.

This is the quantum version of the main property of Einstein's general relativity, where the solutions of the theory are not gravitational field living inside a spacetime, but are themselves defining spacetime.

General relativity provides a set of ten nonlinear partial differential equations for the spacetime metric ( Einstein's field equations ) that must be solved from the distribution of mass-energy and momentum throughout the universe.

The fact that Einstein's field equation is nonlinear is well-known.

Post-Newtonian formalism is a calculational tool that expresses Einstein's ( nonlinear ) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation.

The time differential between two reunited clocks is deduced through purely uniform linear motion considerations, as seen in Einstein's original paper on the subject, as well as in all subsequent derivations of the Lorentz transformations.

Hawking and Ellis attempt to describe the foundation of space itself and its nature of infinite expansion, using differential geometry to examine the consequences of Einstein's General Theory of Relativity.

It enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

Such statements, however, fall in the same category as statements such as " physics assumes that nature applies formulas such as Einstein's E = mc < sup > 2 </ sup > or Newton's F = ma " and " DST models assume that dynamic systems apply differential equations ".

Inspired by Einstein's approach to a unified field theory and Eddington's idea of the affine connection as the sole basis for differential geometric structure for space-time, Erwin Schrödinger from 1940 to 1951 thoroughly investigated pure-affine formulations of generalized gravitational theory.

The thesis, which dealt with problems in differential geometry related to Albert Einstein's theory of relativity, was defended in front of a jury of 11 mathematicians, including Levi-Civita, Vito Volterra, and Guido Castelnuovo.

Synge made outstanding contributions to different fields of work including classical mechanics, general mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry, and Einstein's theory of relativity.