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Schwarzschild metric

The Schwarzschild metric is an exact solution to the of that describes the geometry surrounding a spherically symmetric, non-rotating, uncharged mass in vacuum, such as the exterior region around a star or . It was derived by German physicist in January 1916, mere months after published the complete theory of , making it the first exact non-trivial solution to the equations. In units where the G and c are set to 1, the of the metric takes the form
ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2),
where M is the mass of the central body, r is the radial coordinate, t is the time coordinate, and the angular part corresponds to the metric on a 2-sphere. This metric is asymptotically flat, recovering the Minkowski spacetime of far from the mass (r ≫ M), and it exhibits a coordinate singularity at r = 2M, known as the , which marks the event horizon of a when the mass is sufficiently compact. According to Birkhoff's theorem, established in 1923, the Schwarzschild metric is the unique spherically symmetric solution to the vacuum Einstein equations in four-dimensional , implying that any spherically symmetric evolves to this static form in the exterior region. The solution has profound implications for , underpinning predictions such as , the bending of light by massive bodies (as observed during the 1919 ), and the existence of with inescapable event horizons. For non-vacuum cases inside the mass distribution, the metric can be matched to interior solutions like the Schwarzschild interior solution for a constant-density star.

Historical Development

Schwarzschild's Solution

(1873–1916) was a prominent and known for his contributions to and , including the development of methods for analyzing stellar spectra and the design of optical instruments. As director of the Astrophysical Observatory in Potsdam from 1909, he advanced before enlisting in the during , where he served on the Eastern Front calculating artillery trajectories despite the harsh conditions. In late 1915, while stationed in , Schwarzschild derived the first exact vacuum solution to Einstein's newly formulated field equations for the exterior to a spherically symmetric, non-rotating mass, motivated by the theory's implications for gravitational fields around stars. Schwarzschild completed this work amid personal hardship, solving the complex differential equations during a period of illness that began affecting him on the front lines. He communicated the solution to Einstein via a letter dated December 22, 1915, and the paper was formally submitted to the on January 13, 1916, before being presented by Einstein on the same day and published in the January 1916 issue of Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften. In the original presentation, Schwarzschild expressed the metric in curvature coordinates, where the radial coordinate directly relates to the area of spherical surfaces; this form is the more commonly used one today and exhibits a coordinate at the . Schwarzschild's health deteriorated rapidly due to , a severe autoimmune skin disease likely contracted during his , leading to his being invalided home in March ; he succumbed to the illness on , , at age 42, shortly after completing additional work on stellar interiors. The solution gained further recognition when , a student of , independently derived a similar in his thesis submitted in June , providing an alternative derivation in curvature coordinates and extending applications to particle motion in the field.

Preceding Context

The development of general relativity reached a pivotal moment in November 1915 when presented the final form of the field equations to the on November 25, governing the curvature of in the presence of matter and energy. These equations, expressed as G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, marked the culmination of Einstein's efforts to formulate a theory of consistent with and the . Concurrently, pursued a variational approach to the same equations, delivering a on November 20, 1915, that outlined a unified theory of gravitation and , though his full publication appeared later; historical analysis confirms both arrived at the covariant field equations independently, resolving long-standing debates in favor of parallel discovery. Prior to this breakthrough, Einstein's work on relied heavily on approximate methods, including weak-field expansions that linearized the field equations to recover Newtonian in the low-velocity, weak-curvature . These approximations successfully explained phenomena like the anomalous of Mercury's but fell short for strong-field regimes, prompting the need for exact vacuum solutions to describe outside spherical distributions, such as stars, where the stress-energy tensor vanishes. No non-trivial exact solutions existed at the time, leaving a gap in applying the theory to realistic astronomical sources. The assumption of spherical symmetry in seeking these solutions drew directly from Newtonian gravity's empirical success in modeling planetary and stellar fields, where inverse-square laws held for spherically symmetric bodies like . This conceptual continuity foreshadowed the uniqueness of such vacuum solutions, later formalized by Birkhoff's theorem in 1923, which proved that the spherically symmetric, asymptotically flat metric is unique up to scaling, even for time-dependent cases. Schwarzschild's 1916 derivation would become the first exact realization of this framework.

Mathematical Formulation

Line Element

The Schwarzschild metric provides the line element for the geometry surrounding a spherically symmetric, non-rotating, uncharged M in , serving as the exact to Einstein's field equations in this configuration. In standard (t, r, \theta, \phi), where t is the time coordinate for stationary observers, r is the radial coordinate, and (\theta, \phi) are angular coordinates, the components yield the ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 \, dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin^2 \theta \, d\phi^2\right), adopting the metric signature (-, +, +, +). This expression employs the gravitational constant G and speed of light c, with the time component g_{tt} = -\left(1 - r_s / r\right) incorporating the Schwarzschild radius r_s = 2GM/c^2, which characterizes the scale of strong-field effects near the mass. In natural units where G = c = 1, the line element simplifies to ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin^2 \theta \, d\phi^2\right). The coordinate r functions as the areal radius, such that the geometry of spherical surfaces at fixed r and t has a proper circumference of $2\pi r, preserving the spherical symmetry. The metric's static nature arises from its independence of t, reflecting the absence of time-varying fields or rotation for the central mass. As r \to \infty, the coefficients approach those of the flat Minkowski metric ds^2 = -c^2 dt^2 + dr^2 + r^2 (d\theta^2 + \sin^2 \theta \, d\phi^2), ensuring the solution is asymptotically flat and consistent with special relativity at large distances from the mass.

Derivation from Einstein Equations

The derivation of the Schwarzschild metric begins with the vacuum , R_{\mu\nu} = 0, where R_{\mu\nu} is the tensor, applicable outside the source where the stress-energy tensor vanishes. These equations were formulated by in 1915 as the core of . Given the physical scenario of a spherically symmetric, static (time-independent) , the is assumed to take the general form in curvature coordinates: ds^2 = -e^{2\Phi(r)} dt^2 + e^{2\Lambda(r)} dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2, where \Phi(r) and \Lambda(r) are functions of the radial coordinate r only, and d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2. This respects the spherical symmetry and time independence of the . To solve R_{\mu\nu} = 0, the Ricci tensor components are computed for this metric. The relevant non-zero components yield ordinary differential equations for \Phi and \Lambda. From the \theta\theta-component of the Ricci tensor, one obtains \frac{d}{dr} \left( r e^{-2\Lambda(r)} \right) = 1, which integrates to e^{-2\Lambda(r)} = 1 - \frac{2m(r)}{r}, where m(r) is the mass function representing the enclosed mass up to radius r. The tt- and rr-components of the Ricci tensor then imply that the derivative m'(r) = 0, so m(r) must be constant, m(r) = M, where M is the total mass of the source. Additionally, the tt- and rr-components lead to the key relation \Phi'(r) + \Lambda'(r) = 0, allowing \Phi(r) to be integrated as \Phi(r) = \frac{1}{2} \ln \left(1 - \frac{2M}{r}\right). Thus, e^{2\Phi(r)} = 1 - 2M/r and e^{2\Lambda(r)} = (1 - 2M/r)^{-1}, completing the metric components. This solution is unique for spherically symmetric spacetimes, as established by Birkhoff's theorem, which proves that any such metric must be the Schwarzschild metric (up to a constant rescaling of the time coordinate). The theorem was demonstrated in 1923.

Intrinsic Properties

Symmetries

The Schwarzschild metric exhibits a high degree of , characterized by its and the associated Killing vectors that preserve the . These symmetries arise from the under the assumption of spherical and staticity, making the solution unique up to scaling by the mass parameter. The admits exactly four independent Killing vectors. Outside the event horizon (r > 2M), one is timelike, given by \partial_t, which reflects the static nature of the geometry and allows for a preferred time direction for observers at rest. The remaining three are spacelike, corresponding to rotations around the coordinate axes: \partial_\phi, \cos\phi \partial_\theta - \cot\theta \sin\phi \partial_\phi, and -\sin\phi \partial_\theta - \cot\theta \cos\phi \partial_\phi, generating the full spherical . Inside the horizon, the \partial_t vector becomes spacelike, while the rotational vectors retain their character. This set of four Killing vectors represents the maximum possible for a spherically symmetric solution in four dimensions. The generated by these Killing vectors is \mathbb{R} \times SO(3), comprising continuous time translations along \mathbb{R} and the compact group of spatial rotations SO(3). At spatial , the rotational subgroup aligns with the asymptotic Lorentz of flat Minkowski , but the global isometries lack boosts or translations due to the central mass. These symmetries yield conserved quantities for motion, facilitating the analysis of particle and light paths. The timelike Killing vector \partial_t gives the at E = -u \cdot \partial_t, conserved along any with u^\mu. The rotational Killing vectors produce the components of \mathbf{L}, with L_z = u \cdot \partial_\phi and the other components derived from the full SO(3) action, leading to the conserved magnitude L^2. The maximal number of four Killing vectors ensures the complete integrability of the geodesic equations in this four-dimensional spacetime. This integrability is evident in the separability of the Hamilton-Jacobi equation for the geodesic action, where the symmetries allow the separation of variables into radial, temporal, and angular parts, yielding four independent constants of motion and enabling exact solutions without chaos.

Curvature Tensors

The Schwarzschild metric describes a vacuum region exterior to a spherically symmetric mass, where the Ricci tensor vanishes due to the absence of matter, satisfying the Einstein field equations R_{\mu\nu} = 0 for r > 2M (in units where G = c = 1). This zero Ricci tensor implies that all local contractions of the curvature are zero, leaving the geometry determined solely by the traceless Weyl part. The full captures the spacetime's deviation from flatness, with non-vanishing components reflecting the distortions induced by the central mass. For instance, one key component is R^t{}_{rtr} = -\frac{2M}{r^3} \left(1 - \frac{2M}{r}\right), which quantifies the relative acceleration between nearby geodesics and relates directly to forces on infalling observers. These effects stretch objects radially while compressing them transversely, with strength scaling as M/r^3 far from the horizon. A gauge-invariant measure of the total curvature is the Kretschmann scalar, defined as K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} = \frac{48 M^2}{r^6}. This invariant remains finite at the event horizon (r = 2M, where K = \frac{3}{4 M^4}) but diverges as r \to 0, indicating a true physical singularity only at the center. In vacuum, the Riemann tensor coincides with the Weyl tensor, which for the Schwarzschild geometry is purely electric with respect to static observers, meaning its magnetic part vanishes and all components are proportional to M/r^3. The electric Weyl tensor components, such as \Psi_2 = -M/r^3 in the Newman-Penrose formalism, encode the Coulomb-like gravitational field, confirming the absence of frame-dragging or other magnetic curvature effects. Curvature invariants like the Kretschmann scalar and Weyl-squared demonstrate that the horizon is a coordinate artifact with no intrinsic , while the central divergence signals an unavoidable breakdown of classical .

Coordinate Systems

Standard Schwarzschild Coordinates

The standard Schwarzschild coordinates consist of four parameters: t, which serves as a timelike coordinate asymptotically approaching the proper time measured by stationary observers at spatial infinity; r, the areal radius parameterizing the such that spheres of constant t and r have surface area $4\pi r^2; and \theta, \phi, the conventional polar and azimuthal angles on the unit sphere. These coordinates were introduced in the original solution to Einstein's field equations for a spherically symmetric, non-rotating . The coordinates are valid in the exterior vacuum region where r > 2M, with M denoting the of the central source in geometric units (G = c = 1); they break down at the event horizon r = 2M due to a coordinate singularity and at the r = 0 due to a physical singularity. In this system, the metric component g_{rr} = (1 - 2M/r)^{-1} diverges as r approaches $2M from above, reflecting the pathology of the coordinates at the horizon rather than a true irregularity. Despite this divergence, the proper radial distance along a spacelike path from the horizon to any exterior point r > 2M remains finite, computed as \int_{2M}^{r} \frac{dr'}{\sqrt{1 - 2M/r'}} = \sqrt{r(r - 2M)} + 2M \ln \left( \sqrt{\frac{r}{2M}} + \sqrt{\frac{r - 2M}{2M}} \right), which is finite as r \to 2M^+. The areal radius r does not correspond to the proper radial distance because the spatial line element for radial directions incorporates the factor (1 - 2M/r)^{-1/2}, stretching the measured length relative to \Delta r increasingly near the horizon; this choice prioritizes the geometric area of spherical shells over direct measurement of radial separation. At the horizon, light cones in these coordinates tilt such that future-directed null directions align tangentially, preventing outgoing signals from escaping in finite coordinate time t, yet the system remains well-behaved for freely infalling observers who cross the horizon smoothly and continue to r = 0 without encountering further coordinate issues until the central singularity.

Alternative Coordinate Charts

The standard Schwarzschild coordinates fail to describe the spacetime geometry across the event horizon at r = 2M, where they exhibit a coordinate singularity, necessitating alternative coordinate systems that extend smoothly through this surface while preserving the underlying . These transformations are crucial for analyzing phenomena inside the horizon and for constructing the maximal analytic extension of the Schwarzschild solution. Eddington-Finkelstein coordinates provide a null coordinate extension suitable for ingoing geodesics, replacing the time coordinate t with an advanced null coordinate v = t + r^*, where the tortoise coordinate is defined as r^* = \int^r \frac{dr'}{1 - 2M/r'} = r + 2M \ln \left| \frac{r}{2M} - 1 \right|. In these coordinates, the takes the form ds^2 = -(1 - 2M/r) dv^2 + 2 dv dr + r^2 d\Omega^2, which remains regular at r = 2M, allowing description of the exterior region and the interior. Ingoing null geodesics follow paths of constant v, facilitating the study of infalling matter or light without the pathology of infinite coordinate time in . An outgoing version exists with retarded time u = t - r^*, but it covers only the exterior and regions. Kruskal-Szekeres coordinates offer a maximal extension of the Schwarzschild geometry, covering all causally distinct regions including both interiors and exteriors of black hole and white hole horizons. These are constructed from null coordinates u = t - r^* and v = t + r^* via U = -e^{-u/4M} and V = e^{v/4M} for the right wedge, with the metric ds^2 = \frac{32 M^3}{r} e^{-r/2M} (-dV^2 + dU^2) + r^2 d\Omega^2, where r is implicitly defined by UV = -\left( \frac{r}{2M} - 1 \right) e^{r/2M}. This system removes the coordinate singularity at the horizon and reveals the full structure, including two asymptotically flat regions connected through the Einstein-Rosen bridge, with the Kruskal diagram depicting the maximal extension featuring two horizons and four asymptotic regions. The coordinates are regular everywhere except at the physical curvature singularity r = 0. Painlevé-Gullstrand coordinates describe the Schwarzschild geometry using a timelike coordinate aligned with free-falling observers from , yielding the ds^2 = -dt^2 + \left( dr + \sqrt{\frac{2M}{r}} dt \right)^2 + r^2 d\Omega^2. This form mimics the flow of a test fluid with \sqrt{2M/r} toward the center, extending smoothly through the horizon and into the interior, where radial coordinate r decreases along timelike geodesics of constant angular coordinates. These coordinates highlight the "river model" analogy for spacetimes, useful for visualizing infall without shell-crossing issues. In the black hole interior, null coordinates like Eddington-Finkelstein emphasize light propagation along ingoing paths, whereas free-fall coordinates such as Painlevé-Gullstrand align with massive particle trajectories, providing complementary perspectives: the former suits causal structure analysis, while the latter aids in understanding observer experiences during collapse.

Spatial Embedding (Flamm's Paraboloid)

The spatial embedding of the Schwarzschild geometry provides a visualization of its intrinsic curvature by isometrically mapping a two-dimensional slice into three-dimensional Euclidean space. Specifically, consider the equatorial hypersurface defined by constant time t and \theta = \pi/2, where the induced line element is dl^2 = \frac{dr^2}{1 - \frac{2M}{r}} + r^2 d\phi^2 for r > 2M, with M denoting the mass parameter (in units where G = c = 1). This metric describes a curved surface of revolution, and its embedding exploits the spherical symmetry of the Schwarzschild solution to construct a concrete geometric picture. To achieve the embedding, the surface is placed in cylindrical coordinates (r, \phi, z) within flat , where the line element is dl^2 = dz^2 + dr^2 + r^2 d\phi^2. Matching the metrics requires \left( \frac{dz}{dr} \right)^2 = \frac{1}{1 - \frac{2M}{r}} - 1 = \frac{2M/r}{1 - 2M/r}, yielding \frac{dz}{dr} = \pm \frac{\sqrt{2M/r}}{\sqrt{1 - 2M/r}}. Integrating this from the horizon at r = 2M (where z = 0) gives the embedding function z(r) = 2 \sqrt{2M (r - 2M)} for r \geq 2M. This surface, known as Flamm's paraboloid, forms a of revolution with its narrowest point—a of radius $2M—at the event horizon, flaring outward asymptotically as r increases, resembling a funnel. The illustrates how the spatial outside the horizon deviates from flatness: circumferential distances $2\pi r match those in the , but radial proper distances are stretched due to the , creating the dipped shape. This visualization highlights the wormhole-like structure of the Schwarzschild exterior (without the interior), though it embeds only the asymptotically flat region beyond the horizon. However, such an cannot be extended smoothly into the interior r < 2M using Euclidean space, as the denominator under the square root in dz/dr becomes negative, resulting in imaginary values for z. This impossibility stems from the failure of the interior geometry to satisfy the flare-out condition at the throat, which is necessary for a minimal surface in wormhole spacetimes but is violated in the collapsing-like behavior inside the horizon. Flamm's paraboloid was first constructed by Ludwig Flamm in 1916, shortly after Karl Schwarzschild's solution, as an attempt to interpret the metric's implications for photon orbits and gravitational theory, predating the modern black hole understanding.

Singularities and Black Holes

Coordinate Singularity at the Horizon

In standard Schwarzschild coordinates, the metric exhibits an apparent singularity at the event horizon located at r = 2M, where the time-time component g_{tt} = -(1 - 2M/r) approaches zero and the radial-radial component g_{rr} = (1 - 2M/r)^{-1} diverges to infinity. This behavior indicates a breakdown in the coordinate system, as the metric coefficients become ill-defined, preventing straightforward continuation of geodesics across the horizon in these coordinates. Despite this coordinate pathology, the proper time experienced by an infalling observer remains finite when crossing the horizon. For an observer in free fall near the horizon, the proper time \tau to traverse the horizon from any finite r > 2M is finite, and locally dr/d\tau \approx -1 (in units where c = 1), reflecting that the local geometry allows smooth passage without any physical disruption. In contrast, the coordinate time t required for such an observer to reach the horizon diverges to , meaning that distant observers perceive the infalling object as asymptotically approaching but never quite reaching r = 2M. The removability of this singularity is demonstrated by the finiteness of curvature invariants at r = 2M. Specifically, the Kretschmann scalar, a gauge-invariant measure of curvature given by K = R_{abcd} R^{abcd} = 48 M^2 / r^6, evaluates to K = 3 / (4 M^4) at the horizon, confirming that no intrinsic singularity exists there. This local flatness implies that an observer crossing the horizon encounters no unusual physical effects in their immediate vicinity, though signals attempting to escape to require infinite . The Schwarzschild exterior region (r > 2M) is geodesically complete in the sense that all timelike and geodesics originating outside the horizon remain well-defined without encountering pathologies, but the standard coordinates fail to describe trajectories that cross into r < 2M. This coordinate artifact can be resolved through transformations to alternative charts, such as Kruskal-Szekeres coordinates, which extend the manifold smoothly across the horizon.

Physical Singularity at the Center

The physical singularity at r = 0 in the represents a true pathology of spacetime where the curvature becomes infinite, distinguishing it from coordinate artifacts. This is evidenced by the divergence of scalar curvature invariants, which are independent of coordinate choices. In particular, the , defined as K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}, takes the value K = \frac{48 G^2 M^2}{c^4 r^6} for the and diverges as r \to 0, confirming the presence of a genuine curvature singularity. Similarly, components of the , such as R_{trtr} = -\frac{2GM}{c^2 r^3}, diverge as r \to 0, implying that tidal forces on infalling matter become infinitely strong, stretching objects radially while compressing them transversely. The singularity manifests in the geodesic incompleteness of the spacetime: all future-directed timelike and null geodesics that enter the interior region (r < 2GM/c^2) terminate at r = 0 after a finite affine parameter. For radial timelike geodesics, this incompleteness occurs in finite proper time; specifically, the maximum proper time for an observer falling radially from rest at the event horizon to the singularity is \tau = \pi \frac{GM}{c^3}. This finite proper time underscores that the breakdown is unavoidable for infalling observers, as general relativity fails to describe physics beyond this point. Such geodesic incompleteness is rigorously guaranteed by the singularity theorems of Penrose and Hawking, which apply to the Schwarzschild spacetime due to the presence of trapped surfaces and the focusing of geodesics by gravity. Penrose's 1965 theorem demonstrates that, under conditions like the existence of a trapped surface and energy positivity, timelike geodesics are incomplete, predicting the central singularity in collapse scenarios leading to the Schwarzschild geometry. Although the singularity is physically real and renders geodesics inextendible, it remains hidden behind the event horizon, consistent with the strong cosmic censorship conjecture in classical general relativity, which posits that such pathologies are not visible to distant observers. The Schwarzschild metric, being a vacuum solution to Einstein's equations, is valid only in the exterior region and breaks down at r = 0, where no consistent extension exists within classical general relativity. In realistic astrophysical contexts, such as the collapse of a pressureless dust ball, the interior solution matches onto the Schwarzschild exterior but culminates in the same central singularity, as described in the Oppenheimer-Snyder model. This highlights the universal nature of the physical singularity for spherically symmetric, non-rotating collapse in general relativity.

Interpretation as Black Holes

The event horizon in the Schwarzschild metric, located at the radial coordinate r = 2M (where M is the mass of the central body in geometric units), serves as a one-way boundary for the spacetime, permitting matter and light to cross inward but preventing any escape to the exterior region. This null hypersurface traps all infalling particles and radiation, defining the boundary beyond which causal communication with distant observers is impossible. The Schwarzschild solution, originally an eternal vacuum spacetime, finds physical realization in the collapse of a massive star, as demonstrated by the Oppenheimer-Snyder model of dust collapse. In this 1939 analysis, a spherically symmetric pressureless star implodes under its own gravity, with the exterior region precisely matching the Schwarzschild metric for r > 2M once the event horizon forms, while the interior evolves from a to a . This model illustrates how realistic holes emerge dynamically, with the horizon enveloping the collapsing matter in finite for infalling observers. For eternal, non-rotating black holes, the establishes that the spacetime is uniquely determined by the total mass M alone, excluding any additional "hair" such as multipole moments or other parameters beyond those implied by spherical symmetry. Proven initially by Werner Israel in 1967 for static spacetimes and extended by in the early 1970s to include cases, this theorem underscores the simplicity of Schwarzschild black holes, where all information about the progenitor star is lost except for its mass. The maximally extended Schwarzschild geometry, constructed via Kruskal-Szekeres coordinates, incorporates a region in the past and a parallel exterior connected through the Einstein-Rosen bridge, representing an idealized, eternal structure without formation history. However, realistic black holes formed by stellar collapse correspond only to the future-directed exterior and interior regions beyond the event horizon, excluding the unphysical and bridge components. Modern simulations of non-spinning mergers and collapses confirm the Schwarzschild metric as an excellent approximation for the final state, with the settling rapidly to this vacuum solution outside the horizon after any transient deviations.

Motion

Timelike s (Massive Particles)

The motion of massive test particles in the Schwarzschild follows timelike , which maximize the along the particle's worldline. These can be derived using the from the \mathcal{L} = \frac{1}{2} g_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau}, where \tau is the and the metric g_{\mu\nu} is that of the Schwarzschild solution (assuming units where G = c = 1). Assuming motion in the equatorial plane (\theta = \pi/2), the Euler-Lagrange equations yield the conserved specific energy E = \left(1 - \frac{2M}{r}\right) \frac{dt}{d\tau} and specific angular momentum L = r^2 \frac{d\phi}{d\tau}, arising from the timelike and rotational Killing vectors of the metric. The normalization condition for timelike geodesics, g_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = -1, combines with the conserved quantities to give the radial of motion: \left(\frac{dr}{d\tau}\right)^2 = E^2 - V(r), where the is V(r) = \left(1 - \frac{2M}{r}\right) \left(1 + \frac{L^2}{r^2}\right). This resembles the in Newtonian , with V(r) governing the radial dynamics; for bound orbits, E < 1, and the potential features centrifugal barriers at large r and an attractive gravitational well modified by the spacetime curvature. Circular orbits occur where \frac{dr}{d\tau} = 0 and \frac{d^2 r}{d\tau^2} = 0, requiring E^2 = V(r) and \frac{dV}{dr} = 0. Solving \frac{dV}{dr} = 0 yields the orbital radius r = \frac{L^2}{2M} \left(1 + \sqrt{1 - \frac{12M^2}{L^2}}\right) for stable circular orbits, with corresponding E^2 = \frac{(1 - 2M/r)^2}{1 - 3M/r}. Stability requires \frac{d^2 V}{dr^2} > 0, which holds down to the (ISCO) at r = 6M, where L = \sqrt{12} M and E = \sqrt{8/9}. For r < 6M, no stable circular orbits exist; particles on nearly circular trajectories plunge inward toward the central singularity, as the effective potential lacks a minimum and the radial velocity becomes inexorable. This ISCO marks a fundamental limit on stable accretion and orbital dynamics around a Schwarzschild black hole, with implications for phenomena like X-ray binaries.

Null Geodesics (Photons)

Null geodesics describe the paths followed by massless particles, such as photons, in the Schwarzschild spacetime. Due to the time-translation and rotational symmetries of the metric, these geodesics admit conserved quantities: the specific energy E = (1 - 2M/r) \, dt/d\lambda and the specific angular momentum L = r^2 \, d\phi/d\lambda, where \lambda is an affine parameter. For null geodesics, the condition ds^2 = 0 leads to the radial equation (dr/d\lambda)^2 = E^2 - V_\text{eff}(r), where the effective potential is given by V_\text{eff}(r) = (L^2 / r^2) (1 - 2M/r). This form is obtained by setting the rest mass m = 0 in the general geodesic effective potential, highlighting the absence of a rest energy term for photons. The effective potential features an unstable maximum at the photon sphere radius r = 3M, corresponding to circular null orbits. The impact parameter b = L / E characterizes the asymptotic behavior of incoming photons from infinity. Photons with b < b_c = 3\sqrt{3} M spiral into the event horizon, while those with b > b_c are deflected outward; at b = b_c, they orbit unstably at the photon sphere. This critical value determines the boundary between captured and escaping light rays. In the weak-field limit, far from the central mass, the deflection angle for a photon passing at closest approach r_0 \gg M is \delta\phi = 4M / b, where b \approx r_0. This result, first derived using the Schwarzschild metric, doubles the Newtonian prediction and was confirmed by observations of starlight grazing the Sun during the 1919 solar eclipse. The casts a on distant screens, with angular radius \theta_\text{sh} = b_c / D_A = 3\sqrt{3} M / D_A, where D_A is the to the observer. For Sagittarius A* (M \approx 4 \times 10^6 M_\odot, D_A \approx 8 kpc), the predicted is approximately 50 \muas. The Event Horizon Telescope (EHT) analysis of 2017 observations, released in 2022, imaged this with a measured of $51.8 \pm 2.3 \muas, consistent with the Schwarzschild prediction and providing a direct test of paths near the horizon.