Geodesics in general relativity

• 1. Straight Lines In Curved Space

  ›

  A freely falling particle traces a path through four-dimensional spacetime that physicists call a geodesic. This concept generalizes the idea of a straight line to environments where gravity curves space and time together. The world line of such a particle represents its entire history from start to finish without any external non-gravitational forces acting upon it. Gravity itself is not treated as a force in this framework but rather as a consequence of curved geometry. A planet orbiting a star follows a projection of this four-dimensional curve onto our familiar three-dimensional space. The source of this curvature is the stress-energy tensor which accounts for matter and energy distributions.

• 2. Christoffel Symbols And Motion Equations

  ›

  The full mathematical expression for motion involves Christoffel symbols denoted by the Greek letter Gamma with two lower indices. These symbols function as affine connection coefficients or Levi-Civita connection coefficients within the equation. They are symmetric in their two lower indices and depend on the four spacetime coordinates. The quantity on the left-hand side of the summation represents the acceleration of a particle. This structure mirrors Newton's laws of motion which also provide formulae for calculating acceleration. The Christoffel symbols remain independent of the velocity or other characteristics of the test particle being described. Physicist Steven Weinberg presented a derivation starting from the equivalence principle to establish these relationships. He began by supposing that a free falling particle does not accelerate relative to a freely falling coordinate system near a specific point-event.

• 3. Equivalence Principle Derivation Steps

  ›

  Steven Weinberg outlined a step-by-step process deriving the geodesic equation directly from Einstein's equivalence principle. The first step assumes no local acceleration occurs within a neighborhood of a point-event when using a freely falling coordinate system. Setting the parameter lambda to zero yields an equation locally applicable during free fall. The next step employs the multi-dimensional chain rule to transform derivatives. Differentiating once more with respect to time produces a new relationship involving second derivatives. The left-hand side of this final equation must vanish because of the Equivalence Principle itself. Multiplying both sides by a specific quantity leads to the definition of the affine connection. This definition results in the standard geodesic formula used throughout general relativity. Proper time serves as the local time at a point following the line of motion in question.

• 4. Action Principles And Least Motion

  ›

  The most common technique for deriving the geodesic equation utilizes the action principle and Euler-Lagrange equations. Consider finding a geodesic between two timelike-separated events where the action involves the line element. A negative sign appears inside the square root because the curve must be timelike. Parameterizing this action with respect to a variable allows physicists to vary it against the curve itself. Hamilton's principle dictates that the variation must equal zero to find the path of least action. Integrating by parts and dropping total derivatives simplifies the expression significantly. Multiplying by the inverse metric tensor yields the final geodesic equation. Similar derivations produce analogous results for light-like or space-like separated pairs of points. The goal is to find a curve joining two events which has a stationary interval.

• 5. Field Equation Debates And Singularities

  ›

  Albert Einstein believed the law of motion could be derived solely from field equations for empty space. He argued that generalized laws of motion follow from the condition that fields remain singular nowhere outside generating mass points. One imperfection of the original relativistic theory was its introduction of an independent postulate regarding particle motion. A complete field theory knows only fields and not separate concepts of particles or motion. Both physicists and philosophers have often repeated the assertion that geodesic equations emerge from field equations alone. This claim remains disputed according to David Malament who noted other assumptions are needed. Less controversial is the notion that field equations determine the motion of fluid or dust rather than point-singularities. Tamir published a paper in 2012 titled Proving the principle taking geodesic dynamics too seriously in Einstein's theory.

• 6. Charged Particles And Lorentz Forces

  ›

  Real life particles may carry electric charges causing them to accelerate locally even within free fall. These accelerations occur in accordance with the Lorentz force acting upon the charged test particle. The Minkowski tensor describes this interaction and reduces to the metric tensor in free fall. The resulting equation of motion includes terms representing electromagnetic forces alongside gravitational curvature. Massless particles like photons instead follow null geodesics where the right-hand side equals zero. It is important that these last two equations remain consistent when differentiated with respect to proper time. The letter g with superscripts refers to the inverse of the metric tensor used for raising indices. This final equation does not involve electromagnetic fields directly but applies even as those fields vanish completely.