World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is a circle in space, but its worldline is a helix in spacetime.
For definiteness consider a circular earth orbit (helical world line) of a particle. The particle travels with speed v. An observer on earth sees that length is contracted in the frame of the particle. A measuring stick traveling with the particle appears shorter to the earth observer. Therefore, the circumference of the orbit, which is in the direction of motion appears longer than times the diameter of the orbit.
The 4-acceleration in the earth (non-accelerating) frame is
where is c times the proper time interval measured in the frame of the particle. This is related to the time interval in the Earth's frame by
Here, the 3-acceleration for a circular orbit is
where is the angular velocity of the rotating particle and is the 3-position of the particle.
The magnitude of the 4-velocity is constant. This implies that the 4-acceleration must be perpendicular to the 4-velocity. The 4-acceleration is, in fact, perpendicular to the 4-velocity in this example (see Fermi–Walker transport). The inner product of the 4-acceleration and the 4-velocity is therefore always zero. The inner product is a Lorentz scalar.
It is easily verified that circular orbits satisfy the geodesic equation. The geodesic equation is actually more general. Circular orbits are a particular solution of the equation. Solutions other than circular orbits are permissible and valid.
The geodesic equation in a local coordinate system
Circular orbits at the same radius.
Consider the situation in which there are now two particles in nearby circularpolar orbits of the earth at radius and speed .
The particles execute simple harmonic motion about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their trajectories intersect at the poles.
Imagine we have a spacecraft co-moving with one of the particles. The ceiling of the craft, the direction, coincides with the direction. The front of the craft is in the direction, and the direction is to the left of the craft. The spacecraft is small compared with the size of the orbit so that the local frame is a local Lorentz frame. The 4-separation of the two particles is given by . In the local frame of the spacecraft the geodesic equation is given by
between events 1 and 2 is a minimum. Here S is a scalar and
is known as the Lagrangian density. The Lagrangian density is divided into two parts, the density for the orbiting particle and the density of the gravitational field generated by all other particles including those comprising the earth,
In curved spacetime, the "shortest" world line is that geodesic that minimizes the curvature along the geodesic. The action then is proportional to the curvature of the world line. Since S is a scalar, the scalar curvature is the appropriate measure of curvature. The action for the particle is therefore
where is an unknown constant. This constant will be determined by requiring the theory to reduce to Newton's law of gravitation in the nonrelativistic limit.
The Lagrangian density for the particle is therefore
The action for the particle and the earth is
We find the world line that lies on the surface of the sphere of radius r by varying the metric tensor. Minimization and neglect of terms that disappear on the boundaries, including terms second order in the derivative of g, yields
Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("world line") of a particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance. The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that: * the balls on the world line before/after future/past accelerations are more spaced out due to time dilation. * events which were simultaneous before an acceleration are at different times afterwards (due to the relativity of simultaneity), * events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and * the world line always remains within the future and past light cones of the current event.
Newton's Law of Gravitation in non-relativistic mechanics states that the acceleration on an object of mass due to another object of mass is equal to
For definiteness. consider a particle of mass orbiting in the gravitational field of the earth with mass . The law of gravitation can be written
where is the average mass density inside a sphere of radius .
Gravitational force in terms of the 00 component of the stress–energy tensor
Newton's law can be written
where is the volume of a sphere of radius . The quantity will be recognized from special relativity as the rest energy of the large body, the earth. This is the sum of the rest energies of all the particles that compose earth. The quantity in the parentheses is then the average rest energy density of a sphere of radius about the earth. The gravitational field is proportional to the average energy density within a radius r. This is the 00 component of the stress–energy tensor in relativity for the special case in which all the energy is rest energy. More generally
and is the velocity of particle i making up the earth and in the rest mass of particle i. There are N particles altogether making up the earth.
Relativistic generalization of the energy density
The components of the stress–energy tensor.
There are two simple relativistic entities that reduce to the 00 component of the stress–energy tensor in the nonrelativistic limit
Unfortunately, this acceleration is nonzero for as is required for circular orbits. Since the magnitude of the 4-velocity is constant, it is only the component of the force perpendicular to the 4-velocity that contributes to the acceleration. We must therefore subtract off the component of force parallel to the 4-velocity. This is known as Fermi–Walker transport. In other words,
This is the Einstein field equation that describes curvature of spacetime that results from stress-energy density. This equation, along with the geodesic equation have motivated by the kinetics and dynamics of a particle orbiting the earth in a circular orbit. They are true in general.
Solving the Einstein field equation requires an iterative process. The solution is represented in the metric tensor
Typically there is an initial guess for the tensor. The guess is used to calculate Christoffel symbols, which are used to calculate the curvature. If the Einstein field equation is not satisfied, the process is repeated.
Solutions occur in two forms, vacuum solutions and non-vacuum solutions. A vacuum solution is one in which the stress–energy tensor is zero. The relevant vacuum solution for circular orbits is the Schwarzschild metric. There are also a number of exact solutions that are non-vacuum solutions, solutions in which the stress tensor is non-zero.
Solving the geodesic equations requires knowledge of the metric tensor obtained through the solution of the Einstein field equation. Either the Christoffel symbols or the curvature are calculated from the metric tensor. The geodesic equation is then integrated with the appropriate boundary conditions.
The electromagnetic wave equation is modified from the equation in flat spacetime in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.