Curved spacetime bends the paths of objects and light. Geodesics are the shortest routes through this warped landscape, like cosmic highways for particles and photons.

Free-fall motion follows these geodesics naturally. Understanding geodesics helps us grasp how gravity emerges from spacetime's curvature, a key insight in general relativity.

Geodesics and the Geodesic Equation

Defining Geodesics and Their Properties

• Geodesic represents the shortest path between two points in a curved spacetime
• Geodesics are straight lines in flat spacetime but can be curved in the presence of gravity or spacetime curvature
• Massive particles and light both follow geodesics in spacetime, although massive particles travel at less than the speed of light while light travels along null geodesics
• Geodesics are parameterized by an affine parameter, which is proper time for timelike geodesics and an arbitrary parameter for null geodesics

The Geodesic Equation and Its Components

• Geodesic equation describes the motion of particles in curved spacetime and is derived from the principle of least action
• Geodesic equation is expressed as $\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau} = 0$, where $x^{\mu}$ are the spacetime coordinates, $\tau$ is the proper time, and $\Gamma^{\mu}_{\alpha\beta}$ are the Christoffel symbols
• Christoffel symbols, also known as connection coefficients, describe how the basis vectors of a coordinate system change from point to point in a curved spacetime
• Christoffel symbols are derived from the metric tensor and are used to calculate the curvature of spacetime (Riemann tensor) and the geodesic equation

Worldlines and Their Relation to Geodesics

• Worldline represents the path of an object through spacetime, describing its position in space and time
• Worldlines of freely falling particles in the absence of non-gravitational forces are geodesics in spacetime
• Timelike worldlines represent the paths of massive particles, while null worldlines represent the paths of massless particles (light)
• Worldlines can be used to visualize and analyze the motion of particles in spacetime diagrams (Minkowski diagrams), helping to understand concepts such as time dilation and length contraction

The Principle of Equivalence and Free-Fall

The Principle of Equivalence and Its Implications

• Principle of equivalence states that the effects of gravity are indistinguishable from the effects of acceleration in a small enough region of spacetime
• Einstein's equivalence principle has two parts: the weak equivalence principle (all test particles follow the same trajectory in a gravitational field) and the strong equivalence principle (the laws of physics are the same in all freely falling reference frames)
• Principle of equivalence implies that gravity is not a force but rather a consequence of the curvature of spacetime caused by the presence of mass and energy
• Principle of equivalence leads to the prediction of gravitational time dilation and the bending of light by massive objects

Free-Fall and Its Relation to Geodesics

• Free-fall describes the motion of an object under the influence of gravity alone, without any other forces acting upon it
• Objects in free-fall follow geodesics in spacetime, which are the straightest possible paths in the presence of spacetime curvature
• In a freely falling reference frame, an observer experiences no acceleration and the effects of gravity are nullified (weightlessness)
• Free-fall experiments, such as dropping objects from a height or conducting experiments in a falling elevator, demonstrate the principle of equivalence and the motion along geodesics

Gravitational Time Dilation in Free-Fall

• Gravitational time dilation is a consequence of the principle of equivalence and the curvature of spacetime caused by massive objects
• Clocks in a gravitational field run slower compared to clocks far from the gravitational source, as predicted by the gravitational time dilation formula: $\Delta t = \Delta t_0 \sqrt{1 - \frac{2GM}{rc^2}}$, where $\Delta t$ is the time interval measured by the clock in the gravitational field, $\Delta t_0$ is the time interval measured by the clock far from the gravitational source, $G$ is the gravitational constant, $M$ is the mass of the gravitational source, $r$ is the distance from the center of the gravitational source, and $c$ is the speed of light
• Gravitational time dilation has been confirmed experimentally using atomic clocks at different altitudes (Hafele-Keating experiment) and GPS satellites, which require relativistic corrections to maintain accuracy
• In free-fall, an observer experiences proper time, which is the time measured by a clock that follows the observer's worldline and is unaffected by gravitational time dilation

Proper Time

Definition and Measurement of Proper Time

• Proper time is the time measured by a clock that follows a particular worldline in spacetime
• Proper time is the time experienced by an observer in their own reference frame and is always the maximum time measured between two events
• Proper time is invariant, meaning that all observers agree on the value of proper time along a given worldline, regardless of their relative motion
• Proper time is calculated using the spacetime interval: $d\tau^2 = -ds^2/c^2 = -g_{\mu\nu}dx^{\mu}dx^{\nu}/c^2$, where $d\tau$ is the proper time interval, $ds$ is the spacetime interval, $g_{\mu\nu}$ is the metric tensor, $dx^{\mu}$ and $dx^{\nu}$ are the coordinate differentials, and $c$ is the speed of light

Worldlines and Proper Time

• Worldlines represent the path of an object through spacetime, describing its position in space and time
• Proper time is the time measured along a worldline by a clock that follows that worldline
• Timelike worldlines, which represent the paths of massive particles, always have a positive proper time interval
• Null worldlines, which represent the paths of massless particles (light), have a zero proper time interval, meaning that photons do not experience the passage of time
• Spacelike worldlines, which represent hypothetical paths faster than the speed of light, have an imaginary proper time interval and are not physically realizable

Gravitational Time Dilation and Proper Time

• Gravitational time dilation is the effect where clocks in a gravitational field run slower compared to clocks far from the gravitational source
• Gravitational time dilation is a consequence of the curvature of spacetime caused by massive objects, as described by the principle of equivalence
• Proper time experienced by an observer in a gravitational field is always less than the coordinate time measured by an observer far from the gravitational source
• The difference between proper time and coordinate time depends on the strength of the gravitational field and the distance from the gravitational source, as given by the gravitational time dilation formula: $\Delta \tau = \Delta t \sqrt{1 - \frac{2GM}{rc^2}}$, where $\Delta \tau$ is the proper time interval, $\Delta t$ is the coordinate time interval, $G$ is the gravitational constant, $M$ is the mass of the gravitational source, $r$ is the distance from the center of the gravitational source, and $c$ is the speed of light
• Gravitational time dilation has been confirmed experimentally using atomic clocks at different altitudes (Hafele-Keating experiment) and must be accounted for in GPS satellite systems to maintain accuracy