Part II
Vectors and Tensors
We now begin to develop the mathematics needed for general relativity. General
relativity was invented and developed by Einstein and others using the classic tensor
index calculus invented by nineteenth-century mathematicians such as Riemann and
Ricci and Levi-Cevita. In this approach, vectors are viewed and treated as n-tuples
just as we treated them in special relativity in Part I; the metric tensor is treated as
an n by n array and so forth. We call this the classic or index or component view
of tensors. Most physics applications are still done using the component view. It is
convenient that the component view relies on elementary vector and matrix theory
familiar to all physicists. One important feature of the component view is that vectors
and tensors are fundamentally tied to a coordinate system.
An alternative view was developed later in the twentieth century, and now favored
by many mathematicians and theorists, which we may call the intrinsic or invariant
abstract view. In this view, vectors and tensors and forms are invariant abstract objects
independent of coordinate systems. As an example, one can think of an abstract 3vector in the usual way as an arrow in 3-space. The most important feature of the
abstract view is its independence of a coordinate system; some thus consider it more
physical. The component and abstract views are related simply in that the tensor
components arise as coefficient arrays when the abstract tensor is expanded in a
basis. As such there is a one-to-one correspondence between almost all the concepts
and theorems in the component and abstract views.
The relation between the abstract and component views is somewhat like the relation of classic Greek geometry, using abstract idealized points and lines and curves,
to Cartesian analytic geometry using coordinates and n-tuples. We will discuss most
topics first from the component view and then from the abstract view (Bergmann
1942; Rindler 1969, Weinberg 1972; Adler 1975; Kenyon 1990)
Much of the mathematics in this part is a fairly easy generalization of the vector
calculus used in classical mechanics and electromagnetism, and the 4-vector ideas
of special relativity. Central concepts are those of a Riemann space, vectors and
tensors and forms, affine connections, geodesics that generalize the straight lines of
elementary geometry, and covariant derivatives which generalize the derivatives of
elementary calculus (Lawrie 1990; Arfken 1970). Most of this part is mathematical,
Vectors and Tensors
We now begin to develop the mathematics needed for general relativity. General
relativity was invented and developed by Einstein and others using the classic tensor
index calculus invented by nineteenth-century mathematicians such as Riemann and
Ricci and Levi-Cevita. In this approach, vectors are viewed and treated as n-tuples
just as we treated them in special relativity in Part I; the metric tensor is treated as
an n by n array and so forth. We call this the classic or index or component view
of tensors. Most physics applications are still done using the component view. It is
convenient that the component view relies on elementary vector and matrix theory
familiar to all physicists. One important feature of the component view is that vectors
and tensors are fundamentally tied to a coordinate system.
An alternative view was developed later in the twentieth century, and now favored
by many mathematicians and theorists, which we may call the intrinsic or invariant
abstract view. In this view, vectors and tensors and forms are invariant abstract objects
independent of coordinate systems. As an example, one can think of an abstract 3vector in the usual way as an arrow in 3-space. The most important feature of the
abstract view is its independence of a coordinate system; some thus consider it more
physical. The component and abstract views are related simply in that the tensor
components arise as coefficient arrays when the abstract tensor is expanded in a
basis. As such there is a one-to-one correspondence between almost all the concepts
and theorems in the component and abstract views.
The relation between the abstract and component views is somewhat like the relation of classic Greek geometry, using abstract idealized points and lines and curves,
to Cartesian analytic geometry using coordinates and n-tuples. We will discuss most
topics first from the component view and then from the abstract view (Bergmann
1942; Rindler 1969, Weinberg 1972; Adler 1975; Kenyon 1990)
Much of the mathematics in this part is a fairly easy generalization of the vector
calculus used in classical mechanics and electromagnetism, and the 4-vector ideas
of special relativity. Central concepts are those of a Riemann space, vectors and
tensors and forms, affine connections, geodesics that generalize the straight lines of
elementary geometry, and covariant derivatives which generalize the derivatives of
elementary calculus (Lawrie 1990; Arfken 1970). Most of this part is mathematical,