Non-Inertial Frames of Reference

In spatial geometry we define a point as an entity that has the sole property of having infinitesimal extent in all three dimensions. Analogously, we define an instant as an entity that has infinitesimal extent in time. In Relativity, the geometry of space and time taken together, in both the Special and General theories, we define a locus, the spatio-temporal analogue of a point, as an entity that has the sole property of having infinitesimal extent in all four dimensions. Points and instants constitute the elements of a set that we call a frame of reference, which set fills space or spacetime. A point and an instant provide the locus for the occurrence of an event and events occur in a frame of reference, separated from each other by distance and duration.

The phrase inertial frame of reference denotes a frame that is flat in the same sense that a geometric plane is flat. That statement means that the magnitude of the separation between two events conforms to the number calculated from the four-dimensional analogue of the Pythagorean theorem, Minkowski’s metric equation,

(Eq’n 1)

in which ds has the same value for all observers who measure the separation four-vector (dx,dy,dz,cdt) between the same two events, regardless of where those observers are located in the given inertial frame or in others. In an inertial frame the coordinate axes appear as straight lines. Special Relativity is concerned solely with inertial frames of reference.

The laws of physics take the same form in all inertial frames, according to Special Relativity. In contrast, in a non-inertial frame of reference the laws of physics vary depending on the acceleration of that frame with respect to an inertial frame and the usual physical forces must be supplemented by obligatory forces (usually called fictitious forces). For example, a ball dropped toward the ground does not go exactly straight down, because Earth rotates. Someone rotating with Earth must include the Coriolis force to predict the horizontal component of the ball’s motion. Another example of an obligatory force associated with rotating reference frames is the centrifugal force.

Non-inertial frames of reference provide the subject matter of General Relativity. In the same sense in which inertial frames are flat, non-inertial frames are curved or warped. Algebraically, that fact comes manifest in the deformation of Minkowski’s equation by the inclusion of coefficients that are themselves functions of the coordinates. The best known example of such a deformed metric equation describes Schwarzschild’s solution of Einstein’s equation,

(Eq’n 2)

which we must express in polar coordinates because it describes the spacetime enveloping a uniform, spherically symmetric body of mass M.

The Curvature of Space and Time

We have a 4x4 matrix that contains the coefficients of the metric equation and we call it the metric tensor. In the case of the Schwarzschild metric we have that tensor as

(Eq’n 3)

With that tensor we can represent the metric equation as

(Eq’n 4)

in which we follow Einstein’s convention of automatically summing over
repeated indices. Note that the generalized coordinates q^{i} are
represented as contravariant vectors: we treat all measured quantities as
contravariant. We also have a contravariant version of the metric tensor, which
conforms to the requirement that

(Eq’n 5)

the Kronecker delta, which in matrix form is the identity matrix. We can obtain the contravariant metric tensor from its covariant counterpart by calculating

(Eq’n 6)

The cofactor is simply the determinant of the matrix left over when we delete
the i-th row and the k-th column from g_{ik}.

The metric tensor has derivatives and those derivatives appear in several functions that tell us about the shape and evolution of spacetime and of the motions of objects embedded in spacetime.

The first of those functions is the Christoffel symbol

(Eq’n 7)

If we have a vector v^{p}=dq^{p}/dτ,
in which τ
represents a parameter (such as proper time), then we also have

(Eq’n 8)

(See "A Little Tensor
Geometry" for the derivation). That equation enables us to calculate the
acceleration due to gravity from a knowledge of the metric tensor that describes
spacetime deformed by the presence of a massive body. The Christoffel symbol
also participates in the covariant derivative of a covariant vector (v_{i}=g_{ik}v^{k}),

(Eq’n 9)

The first term on the right represents the partial derivative that we would get if we were differentiating with respect to rectilinear coordinates and the second term modifies that result to account for the curvature of the coordinates.

The metric tensor also participates in Riemann curvature tensor, which is a function of the Christoffel symbol,

(Eq’n 10)

That tensor comes from a double covariant derivative,

(Eq’n 11)

and it gives us something analogous to the curl function of vector analysis.

Geodesic deviation gives us the primary application of the Riemann curvature tensor. Imagine single points tracing out geodesic lines in accordance with some parameter τ. Pick two geodesics that run close to each other and represent the distance between their two generating points as

(Eq’n 12)

The two generating points conform to Equation 8 and from that fact we can infer the equation of geodesic deviation,

(Eq’n 13)

The three vectors on the right side of that equation
correspond to three sides of a geometric solid whose volume is changing. If we
multiply Equation 13 by dτ^{2}
and divide it by dz^{i} (with
ξ^{m}=dz^{m}),
then we have

(Eq’n 14)

(In that equation x, y, and z do not represent the standard Cartesian
coordinates, but represent the generalized curvilinear coordinates in a way that
avoids confusion.) If dx, dy, and dz represent unit vectors, then we have their
product as the volume of a deformed unit cube. On the left side we have the
differential change in the rate of change in dz^{i} with respect to the
incrementing or decrementing of dz^{i} and that corresponds to a change
in the rate at which the solid’s volume is changing.

We can use Equation 14 to compare the volume of a
minuscule element of space dV_{R} to the volume that the same
differential geometric solid would possess in Minkowski space dV_{M}. To
do that we need to calculate Equation 14 in all four directions of spacetime.
But that calculation automatically permutes the variables x, y, and z in a way
that makes the result six times as large as it should be. Compensating for that
fact, we have

(Eq’n 15)

That equation looks like something that we could simplify easily by eliminating the z-component. All we need to do is to convert the i-index into the m-index, sum the result, and write

(Eq’n 16)

in which the Ricci tensor

(Eq’n 17)

comes out as a matrix representing the trace of the Riemann curvature tensor. In calculating that trace, we replaced 256 elements with 16 elements. The Ricci tensor thus gives us a kind of average of the Riemannian curvature of the given region of spacetime.

We can take yet another step in contracting our geometric
tensor. Just as the product g^{ik}g_{ki}=4 gives us a measure of
the dimensionality of the spacetime described by the metric tensor, so the
product

(Eq’n 18)

gives us a measure of the scalar curvature (represented by the Ricci scalar R) of the spacetime that the metric tensor describes. To be more specific, we say that the Ricci scalar represents the degree to which the volume of a given differential geometric solid in a curved, Riemannian space differs from the volume of the same figure in Euclidean space.

The Bianchi Identities

We want to examine some of the symmetries of the Riemann tensor. In order to do that we must convert the contravariant index on the tensor to a covariant one. Thus we get

(Eq’n 19)

With that equation we find that we can permute the last three indices of the Riemann tensor and thereby obtain the first of the Bianchi identities,

(Eq’n 20)

Taking the covariant derivative of the Riemann tensor,

(Eq’n 21)

enables us to devise the second Bianchi identity,

(Eq’n 22)

In this case we have permuted the indices c, d, and f.

Equation 19 also enables us to deduce three more relations involving the Riemann tensor. That deduction involves merely interchanging the first pair of indices, doing the same with the second pair of indices, and then interchanging the pairs. We thus get

(Eq’ns 23)

Those equations, along with Equation 20, tell us that the Riemann tensor contains a lot of empty space and a lot of redundancy. In four dimensions the tensor has 256 elements. Only a few of those elements can be non-zero and independent of the values of the other elements.

The first of Equations 23 tells us that all elements for which a=b must equal zero, the only definite number that’s equal to its own negative. Thus we keep only the elements for which ab=12, 13, 14, 23, 24, 34, noting that the indices in which the digits are reversed from those have the same values and may be discarded as redundant. What we have left consists of six times the number of elements denoted by the indices cd. The second of Equations 23 tells us that we must dismiss all elements for which c=d, so we have only the elements for which cd=12, 13, 14, 23, 24, 34. We have thus dismissed all but 36 elements of the Riemann tensor (as zero valued) and we can display them in a 6x6 matrix,

ab\cd |
12 |
13 |
14 |
23 |
24 |
34 |

12 |
R |
R |
R |
R |
R |
R |

13 |
R |
R |
R |
R |
R |
R |

14 |
R |
R |
R |
R |
R |
R |

23 |
R |
R |
R |
R |
R |
R |

24 |
R |
R |
R |
R |
R |
R |

34 |
R |
R |
R |
R |
R |
R |

That matrix divides into three main parts. The primary diagonal runs from the upper left corner to the lower right corner (ab=cd). The lower triangle consists of the elements that lie below and to the left of the primary diagonal and the upper triangle consists of the elements that lie above and to the right of the primary diagonal. The third of Equations 23 tells us that the two triangles are mirror images of each other, so we can discard one of them (say, the one on the lower left) as dependent and, therefore, irrelevant. That leaves us with 21 presumably non-zero, independent elements of the Riemann tensor.

Equation 20 tells us that one more element in the matrix is redundant. If we select an element that has two digits identical to each other in its index, the equation tells us nothing new: for example, the index 1213 gives us

(Eq’n 24)

That equation merely tells us that R_{1213}=R_{1312}, which
we already knew. We can only get new information from Equation 20 if all of the
digits in the index differ from one another. So we have

(Eq’n 25)

That equation tells us that we can calculate the value of one element (R_{1324})
from the sum of two other elements in our matrix (R_{1234} and R_{1423}),
so it’s redundant and we dismiss it. We thus determine that the Riemann tensor
has 20 elements that are independent of all of the other elements, 20 elements
that are both necessary and sufficient to tell us how space and time are
deformed away from a flat Minkowskian spacetime.

Just as the Christoffel symbols tell us how fast the coordinates turn, so the Riemann curvature tensor tells us about the tidal forces that act on a rigid body moving along a geodesic line. It tells us more directly about the coordinate system and indirectly about the spacetime to which the coordinates conform. Basically the Christoffel symbols represent the gravitational forcefield, while the metric tensor represents the corresponding gravitational potential, so certainly the Riemann tensor tells us how the gravitational field varies and generates tidal forces.

We can simplify our treatment of spacetime curvature by organizing the independent elements of the Riemann tensor into two simpler tensors – the Ricci tensor and the Weyl tensor. Just as the metric tensor encodes the length of the 4-distance between two events, so the Ricci tensor encodes the change in the volume of a spherical region whose center moves along a geodesic path while its other points follow nearby geodesics. The Weyl tensor tells us how that sphere changes shape, to or from an ellipsoid for example. The values of the elements of the Ricci tensor are determined, via Einstein’s equation, by the local distribution of mass-energy and momentum. The values of the elements of the Weyl tensor are determined by indirect sources, such as gravitational waves and the tidal distortions encoded in the metric tensor. In this simple essay the Weyl tensor is irrelevant and I won’t discuss it further.

Most simply put, the Ricci curvature tensor tells us the proportionate amount by which the volume of a minuscule spherical region of space following a geodesic on a Riemannian manifold differs from the volume of the same region in Euclidean space. In terms of the Ricci tensor Equation 16 tells us

(Eq’n 16)

To obtain the Ricci tensor we merely contract the Riemann tensor on its first and third indices, as in Equation 17,

(Eq’n 26)

Equations 23 tell us that this is the only contraction that we can use. If we
contract R_{abcd} on its first and second indices or on its third and
fourth indices, the antisymmetry of the tensor yields a null matrix (a matrix
whose elements all equal zero). If we contract the Riemann tensor on the first
and fourth indices or on the second and third indices, we get -R_{bd},
which is essentially what we have in Equation 26, except for the minus sign.

We can also contract the Ricci tensor to get the Ricci scalar,

(Eq’n 18)

In that calculation we get 16 partial products (g^{11}R_{11}+g^{12}R_{12}+....),
which we then sum to get the Ricci scalar.

To interpret Einstein’s Equation we imagine a minuscule
cube of space and examine what happens to its volume as it follows a geodesic
path through warped spacetime. Look again at Equation 13 and imagine that î^{i}
represents the width of the cube in the i-th direction. If we multiply that xi
by the area of one of the two faces that it penetrates, we thereby calculate the
volume of the cube. Multiply the values of xi in Equation 13 by the appropriate
facial areas and then divide the equation by the volume of the cube (which is
the same in all three calculations on the right-hand side) and we get

(Eq’n 27)

in which we have the velocities expressed as fractions of the speed of light.

I’ll note in passing that there’s a minus sign on the right side of that equation in many presentations. I have a plus sign because I’m using a timelike signature on the metric equation where others use a spacelike signature; that is, in the Minkowski metric the diagonal elements of the matrix are (1, 1, 1, -1) in my work and (1, -1, -1, -1) in that of others.

If we divide a region of spacetime into small enough
patches, we can treat each patch as indistinguishable from an inertial frame of
reference. In the local inertial coordinates of such a patch we have â^{i}=(0,
0, 0, 1) at ô=0,
when our microcube floats at rest and Equation 27 tells us that

(Eq’n 28)

which tells us how our microcube begins changing its volume.

Einstein’s Field Equation

In Newtonian gravitational theory Poisson’s equation relates the gravitational potential in a region of space to the density of mass of the matter occupying that space,

(Eq’n 29)

in which G represents the Newtonian gravitational constant. Because the negative gradient of the potential corresponds to the associated forcefield, we also have that equation in the form of Gauss’s equation,

(Eq’n 30)

or, in integral form,

(Eq’n 31)

in which integrating the field **g** over the surface (∂V)
bounding the volume V stands in proportion to the total mass within that volume.

(Unfortunately, physicists use the lower-case gee to represent both the acceleration due to gravity and the determinant of the metric tensor, both of which appear in General Relativity. But because we must represent gravitational acceleration as a vector, while the determinant is a scalar, we can avoid confusing the two gees by boldfacing (or drawing an arrow over) the one representing the vector quantity.)

Einstein intended to create a more general, more detailed version of Poisson’s equation, one that would explicitly acknowledge the theory of Relativity. He proceeded by deducing the equation he wanted from the principle of least action; that is, he wrote

(Eq’n 32)

and asserted that the variation of that action with respect to a variation in the metric tensor equals zero. In mathematical paraphrase, we have

(Eq’n 33)

Because the variation of the metric tensor is purely arbitrary, the quantity enclosed in the square brackets necessarily equals a perfect zero and that fact gives us the analogue of the Euler-Lagrange equations,

(Eq’n 34)

in which T^{ik} represents the stress-energy tensor (written in
contravariant form because it contains measured quantities) and

(Eq’n 35)

As shown in other essays, we know that

(Eq’ns 36)

so we have Equation 34 as

(Eq’n 37)

which is Einstein’s Equation, describing the evolution of the metric due to the presence of stress-energy.

Using the timelike sign convention in the metric signature (+, +, +, -) on indices 1, 2, 3, 4, we have the stress-energy tensor as

(Eq’n 38)

In that matrix matter and radiation are treated as fluids and appear through
their energy and momentum densities. Each component of T^{ik} represents
the amount of the i-component of the linear-momentum density being moved in the
k-direction (think of a flowing river being shifted sideways). The T^{44}
term equals the mass-energy density of the matter and radiation plus the fluid’s
pressure (which encodes the kinetic energies of the particles as an energy
density). The terms T^{11}, T^{22}, and T^{33} each
equal the fluid’s pressure plus the fluid’s overall kinetic energy density
expressed in the indicated direction,

(Eq’n 39)

The terms T^{21}, T^{31}, T^{32}, T^{12}, T^{13},
and T^{23} denumerate the shear stresses in the fluid; that is, the
transfer of linear momentum laterally. The terms T^{14}, T^{24},
and T^{34} denumerate the components of the fluid’s net linear momentum
density being transferred in the temporal direction, which means that they
simply represent the fluid’s momentum density. Finally the terms T^{41},
T^{42}, and T^{43} tell us how much timeward momentum (that is,
energy) gets transferred into the three spatial dimensions and that, of course,
is simply the momentum density again.

Recalling that R=g^{mn}R_{nm}, we can
raise the indices of the Ricci tensor and lower the indices of the metric tensor
by the standard technique to get R=g_{ik}R^{ki}. If we multiply
Equation 37 by the covariant metric tensor, we get

(Eq’n 40)

In going from the first line to the second I made use of the fact that g_{ik}g^{ki}=4.
Substituting R=2êT into Equation 37 and rearranging gives us

(Eq’n 41)

Since T merely equals the trace of the stress-energy tensor, we have in consequence

(Eq’n 42)

the latter equality coming from Equation 28.

If we plot a spherical array of points initially at rest around a spherically symmetric body of mass M, then we have Equation 42 as

(Eq’n 43)

We assume that the pressures have ignorable values relative to the mass-energy density, so we can delete the three pressure terms from that equation. Further, the equation only has non-zero values inside the body of mass M, so we focus our attention on that location.

Both sides of the equation denote constants, the one on the left following necessarily from the constancy of the term on the right side of the equality sign. That fact allows us to carry out a simple double multiplication by and integration with respect to the differential proper time and thereby obtain

(Eq’n 44)

Multiplying that equation through by the volume gives us

(Eq’n 45)

As our array of points shrinks, the volume that they occupy will change by

(Eq’n 46)

We thus calculate

(Eq’n 47)

Letting the deltas become differentials and replacing kappa with its explicit form (Equation 35) gives us

(Eq’n 48)

which describes Newton’s law of gravity.

Thus we see that the warping of spacetime by the presence of mass-energy causes space itself to accelerate and to carry along any objects residing in it. This demonstrates what we mean by a non-inertial frame of reference: it’s a frame in which an unforced body accelerates nonetheless.

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