where a sum over μ is performed according to the rules of Einstein notation.

The stress – energy tensor involves the use of superscripted variables which are not exponents ( see tensor index notation and Einstein summation notation ).

More compactly, using the Einstein summation notation, del is written as

which is written as a < sub > i </ sub > b < sup > i </ sup > in Einstein notation.

Thus, in mathematical physics, the structure coefficients are often written c < sub > i, j </ sub >< sup > k </ sup >, and their defining rule is written using the Einstein notation as

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.

# REDIRECT Einstein notation

# REDIRECT Einstein notation

# REDIRECT Einstein notation

# REDIRECT Einstein notation

In the Einstein notation this summation is built into the notation.

* Einstein notation

In Einstein notation, contravariant components are denoted with upper indices as in

In Einstein notation, covariant components are denoted with lower indices as in

See Einstein notation for details.

The valence of a tensor is the number of variant and covariant terms, and in Einstein notation, covariant components have lower indices, while contravariant components have upper indices.

* Einstein notation

where we are now using Einstein notation for the tensors, in which repeated indices in a product are summed.

Note that the μ and ν summation indices range from 0 to 3: see Einstein notation.

; Einstein notation

Given a local smooth frame ( e < sub > 1 </ sub >, …, e < sub > k </ sub >) of E over U, any section σ of E can be written as ( Einstein notation assumed ).

this means that the covariant divergence of the Einstein tensor vanishes.

Using the Einstein summation convention, with regular ( contravariant ) vectors having upper indices, this is computing

( Here, the Einstein summation convention is used, and the vector fields are thought of as first order linear differential operators, and the components are often called contravariant components.

The no-hair theorem was originally formulated for black holes within the context of a four-dimensional spacetime, obeying the Einstein field equation of general relativity with zero cosmological constant, in the presence of electromagnetic fields, or optionally other fields such as scalar fields and massive vector fields ( Proca fields, spinor fields, etc.

The crucial new idea, for Einstein, was the introduction of a tetrad field, i. e., a set of four vector fields defined on all of such that for every the set is a basis of, where denotes the fiber over of the tangent vector bundle.

* A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian – Einstein metric ( Donaldson 1987b ).

( a timelike unit vector field and three spacelike unit vector fields, respectively ) can always be found in which the Einstein tensor has a particularly simple appearance:

In connection with this, it may be shown that every Killing vector field is a matter collineation ( by the Einstein field equations, with or without cosmological constant ).

For example, the Schwarzschild solution has a Killing algebra of dimension 4 ( 3 spatial rotational vector fields and a time translation ), whereas the Friedmann-Lemaitre-Robertson-Walker ( FLRW ) metric ( excluding the Einstein static subcase ) has a Killing algebra of dimension 6 ( 3 translations and 3 rotations ).

In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

Suppose that there are two reference frames S and S < nowiki > '</ nowiki >, where S < nowiki > '</ nowiki > is moving relative to S. The origin of S < nowiki > '</ nowiki > moves along some curve in S, which can be traced out by some vector C which is a function of t. The Einstein force is the apparent force acting on a particle of mass m in the S < nowiki > '</ nowiki > frame, and is defined by

For the specific heats at least, the limiting value itself is definitely zero, as borne out by experiments to below 10 K. Even the less detailed Einstein model shows this curious drop in specific heats.

A Bose – Einstein condensate ( BEC ) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero.

Note that the above formula is only applicable to classical ideal gases and not Bose – Einstein or Fermi gases.

Albert Lawrence Brooks ( born Albert Lawrence Einstein ; July 22, 1947 ) is an American actor, voice actor, writer, comedian, and director.

From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.

The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path.

A Bose – Einstein condensate ( BEC ) is a state of matter of a dilute gas of bosons cooled to temperatures very near absolute zero ( or ).

The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by Bose – Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons.

This solution had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite.

This so-called Einstein model is, however, unstable to small perturbations — it will eventually start expanding or contracting.

The universe described by the Einstein model is static ; space is finite and unbounded ( analogous to the surface of a sphere, which has a finite area but no edges ).

The Institute launched the Kavli Nanoscience Institute at Caltech in 2006, the Keck Institute for Space Studies in 2008, and is also the current home for the Einstein Papers Project.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

* 1932 – German-born Swiss physicist Albert Einstein is granted an American visa.

This is sometimes known as the Planck – Einstein equation.

After important contributions of Hendrik Lorentz and Henri Poincaré, in 1905, Albert Einstein solved the problem with the introduction of special relativity, which replaces classical kinematics with a new theory of kinematics that is compatible with classical electromagnetism.

Einstein struggled to the end of his life for a theory that could better comply with his idea of causality, protesting against the view that there exists no objective physical reality other than that which is revealed through measurement interpreted in terms of quantum mechanical formalism.

The most prominent opponent of the Copenhagen interpretation is Albert Einstein.

Einstein maintained that quantum mechanics is physically incomplete and logically unsatisfactory.

According to Karl Popper these experiments showed that the class of " hidden variables " Einstein believed in is erroneous.

In astrophysics, gravitational redshift or Einstein shift is the process by which electromagnetic radiation originating from a source that is in gravitational field is reduced in frequency, or redshifted, when observed in a region of a weaker gravitational field.