Geodesics in general relativity

• 1. Introduction

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  Geodesics in general relativity offer a way to see the entire universe differently. A planet does not follow a curved path around a star because gravity pulls it sideways. It follows the straightest possible line through a curved spacetime, and that line happens to look, from our three-dimensional vantage point, like an orbit. This is the heart of Einstein's theory, and it rewrites the meaning of motion itself.

  The question at the center of this story is deceptively simple: what does it mean to travel in a straight line when space and time are not flat? The answer has occupied physicists and mathematicians for over a century, and it sits at the boundary between geometry and physics, between the abstract and the everyday. Everything floating inside the International Space Station follows a geodesic. So does the light bending around a massive galaxy. So, in a sense, does anyone who has ever fallen.

  Albert Einstein himself wrestled with whether geodesic motion was a truly independent postulate of his theory, or whether it could be derived from deeper principles. That unresolved tension, between the field and the particle, the geometry and the matter, runs through every derivation that follows.

• 2. Geometry Instead of Force

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  Gravity in general relativity is not a force at all. It is a consequence of curved spacetime, and the source of that curvature is the stress-energy tensor, which represents matter and energy. A particle free from all non-gravitational, external forces moves not because something pushes or pulls it, but because spacetime itself is shaped in a way that guides it.

  The path of a planet orbiting a star is, from this viewpoint, the projection of a four-dimensional geodesic down into three-dimensional space. The orbit looks curved in space, but it is as straight as anything can be in the four-dimensional geometry that includes time. The planet is not being deflected; it is coasting.

  This reframing carries a striking implication for how acceleration reads in the theory. When the geodesic equation is written using coordinate time instead of proper time, it reduces, for slow-moving particles, to a form that says all test particles at the same place and time undergo the same acceleration. That is a well-known feature of Newtonian gravity, and the fact that general relativity recovers it in the slow-speed limit was crucial evidence that the new theory was consistent with centuries of observation.

• 3. The Equation and Its Symbols

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  The geodesic equation carries, on its left-hand side, a quantity that represents the acceleration of a particle. In that sense it is directly analogous to Newton's laws of motion, which also provide a formula for acceleration. The analogy goes only so far, though, because the equation is written in the language of curved geometry.

  The objects called Christoffel symbols appear throughout the equation. They are also known as the affine connection coefficients or the Levi-Civita connection coefficients, and they are symmetric in their two lower indices. Critically, these symbols are functions only of the four spacetime coordinates. They do not depend on the velocity, acceleration, or any other property of the particle whose motion the equation describes.

  Greek indices in the equation take the values 0, 1, 2, and 3, representing the four dimensions of spacetime. The scalar parameter written as s can stand for the proper time, which is the time measured by a clock moving along the geodesic. Writing the equation in terms of coordinate time instead of proper time is mathematically equivalent, and that coordinate-time form turns out to be especially useful for computational work and for drawing direct comparisons between general relativity and Newtonian gravity.

• 4. Paths to the Same Equation

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  Physicists have found multiple independent routes to the geodesic equation, and each one illuminates a different facet of the same geometry. Physicist Steven Weinberg derived the equation directly from the equivalence principle, beginning from the premise that a freely falling particle does not accelerate relative to a freely falling coordinate system. From that starting point, applying the multi-dimensional chain rule twice and defining the affine connection in Weinberg's own formulation, the geodesic equation follows.

  A second route runs through the action principle, which is the most common technique in modern physics. One constructs an action using the line element, with a negative sign inside the square root to ensure the curve is timelike, then varies the action with respect to the curve. Applying the Euler-Lagrange equation and multiplying by the inverse metric tensor yields the geodesic equation, with the Christoffel symbols now defined explicitly in terms of the metric tensor. Analogous derivations, with minor amendments, handle geodesics between lightlike or spacelike separated pairs of points.

  A third derivation comes from the autoparallel transport of curves. This approach draws on lectures given by Frederic P. Schuller at the We-Heraeus International Winter School on Gravity and Light. It begins by defining a smooth manifold with a connection and describing a curve that is autoparallelly transported, meaning the curve carries its own tangent vector along itself without twisting. Applying the Leibniz rule and the linearity of the connection, renaming dummy indices, and simplifying leads once again to the same geodesic equation.

• 5. Einstein's Unfinished Question

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  Einstein held a strong conviction that the geodesic equation should not be an independent postulate sitting alongside his field equations. He wrote that one of the imperfections of the original relativistic theory of gravitation was precisely that it introduced the law of geodesic motion as a separate assumption. A complete field theory, he argued, should know only fields and not the independent concepts of particle and motion.

  He believed it could be shown that the law of motion, generalized even to the case of arbitrarily large gravitating masses, could be derived from the field equations for empty space alone, specifically from the vanishing of the Ricci curvature. The condition, in his framing, was that the field be singular nowhere outside its generating mass points.

  David Malament later offered a more cautious assessment. Though the geodesic principle can be recovered as a theorem within general relativity, Malament wrote, it is not a consequence of Einstein's equation or the conservation principle alone. Other assumptions are required to derive the relevant theorems. The motion of a fluid or dust is less controversial in this regard: the field equations do determine it. The harder case, the motion of a point singularity, remains disputed territory.

• 6. Charged Particles and Null Geodesics

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  The derivation from the equivalence principle rests on the assumption that particles in a local inertial frame are not accelerating. Real particles, however, may carry electric charge, and a charged particle in an electromagnetic field does accelerate locally in accordance with the Lorentz force. This changes the starting point of the derivation.

  By incorporating the Minkowski tensor into the starting equations and then introducing the metric tensor, which is symmetric and locally reduces to the Minkowski tensor in free fall, the resulting equation of motion describes a particle moving along a timelike geodesic. A separate condition, tied to the Christoffel symbols and the metric tensor, ensures internal consistency when the equation is differentiated with respect to proper time.

  Massless particles such as photons do not follow timelike geodesics. They follow null geodesics, which are obtained by replacing the value on the right-hand side of the relevant equation with zero rather than negative one. The metric tensor's inverse, denoted by the letter g with superscripts, is used throughout general relativity to raise and lower tensor indices, a procedure that connects the mathematics of the metric to the physical quantities the equations describe.

• 7. Stationary Intervals and Curved Spacetime

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  A geodesic between two events in spacetime can also be described as the curve with a stationary interval, a four-dimensional length. Stationary here is used in the sense from the calculus of variations: the interval along the geodesic varies minimally among nearby curves.

  In simply connected Minkowski space, exactly one geodesic connects any given pair of events. For a timelike geodesic in that flat spacetime, this is the curve with the longest proper time between the two events. Curved spacetime, though, allows a pair of widely separated events to have more than one timelike geodesic between them, and the proper times along those different geodesics are not in general equal. For some of those geodesics, a nearby curve connecting the same two events can have either a longer or a shorter proper time.

  For spacelike geodesics, the situation differs again. In Minkowski space a spacelike geodesic is a straight line, but any nearby curve that differs from it purely spatially will have a longer proper length, while a nearby curve that differs from it purely temporally will have a shorter one. These distinctions matter practically: choosing the right kind of parameter, called an affine parameter, is what makes the right-hand side of the equation of motion vanish and yields the clean final form of the geodesic equation.