The standard cosmological model is based on an equation published by Einstein in 1915, which the brilliant physicist would generalize two years later (see FOCUS) for ambiguous reasons. It describes how matter and energy in the universe shape and alter the geometry of spacetime, introducing a new paradigm: gravity is no longer a force between two or more bodies, but a geometric property of space, distorted by the presence of matter and energy. Thus, the curvature of space near matter is seen as the effect of the gravitational field generated by that matter. Einstein's gravitational field equation is a partial differential equation involving some curious and fascinating mathematical objects: tensors.
A little (linear) algebra
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A tensor is an abstract object encountered in linear algebra and differential geometry. We begin with a vector space V over a commutative field K. A field is a set of numbers equipped with two fundamental operations (addition and multiplication) and possessing certain "pleasant" properties (an identity element denoted by 0 for addition, an identity element denoted by 1 for multiplication, associativity of these operations, and so on). Vectors (the elements of the set called the vector space) and scalars (the elements of the field) interact through scalar multiplication, which produces other vectors.
Vectors also interact through vector addition, which likewise produces new vectors. Using these two operations together, we can construct any number of vectors from others by taking linear combinations. A set E of vectors that spans V and whose elements are independent is called a basis of the vector space. Every basis has the same number of elements, and this number defines the dimension of the space.
The dimension may even be infinite! If V has finite dimension n and E is a basis of V consisting of the vectors e1, e2… *en*, then every vector v in V can be expressed as the following linear combination:
