general relativity equations

From: veritasium

The general theory of relativity is currently our best theory of gravity [01:32:00]. It arose, at least in part, due to a fundamental flaw in Newtonian gravity [01:57:00].

From Newton’s Absurdity to Einstein’s Curvature

In the 1600s, Isaac Newton developed his theory of gravity, concluding that every object with mass attracts every other object [02:02:00]. However, Newton was troubled by the idea of masses exerting force on each other across vast distances without any mediation, describing it as “so great an absurdity” [02:14:00].

Over 200 years later, Albert Einstein resolved this issue [02:42:00]. Rather than direct forces, Einstein’s theory of general relativity posits that a mass, such as the Sun, curves the spacetime in its vicinity [02:48:00]. This curvature then propagates, influencing other masses [02:58:00]. For example, the Earth orbits the Sun because the spacetime it travels through is curved [03:03:00]. This means that masses are affected by the local curvature of spacetime, eliminating the need for “action at a distance” [03:09:00].

Einstein’s Field Equations

Mathematically, this concept is described by Einstein’s field equations [03:16:00]. These equations link the distribution of matter and energy on one side to the resultant curvature of spacetime on the other [03:30:00]. Although appearing as a single line, they are actually a family of coupled differential equations, making them complex to solve [03:43:00].

Understanding Spacetime with Light Cones

To understand solutions to Einstein’s equations, we can use light cones in spacetime diagrams [04:07:00].

• Future Light Cone: If a flash of light goes off, it creates a “light bubble” within which all future events for an observer must occur, as nothing can travel faster than light [04:14:00]. When time is plotted up the screen, this bubble traces out a cone [04:35:00].
• Past Light Cone: Similarly, light converging from all corners of the universe defines a past light cone, encompassing all events that could have affected an observer up to the present moment [05:08:00].

In these diagrams, light rays always travel at 45 degrees when axes are scaled conventionally [04:43:00].

The “spacetime interval” is used to measure the distance between two events in spacetime [05:31:00]. For flat spacetime, the formula is uniform [05:41:00]. However, around a mass, spacetime is curved, requiring a modified equation to account for the geometry [05:54:00]. Solutions to Einstein’s equations provide this modification, describing how spacetime curves and how to measure separations within that curved geometry [06:00:00].

Schwarzschild Solution: The First Black Hole

Albert Einstein published his equations in 1915, but he couldn’t find an exact solution [06:12:00]. Karl Schwarzschild, an astrophysicist stationed on the Eastern Front during WWI, found the first non-trivial solution within weeks [06:19:00].

Schwarzschild simplified the scenario to an eternal, static universe with only a single, spherically symmetric, electrically neutral, non-rotating point mass [06:45:00]. His “Schwarzschild metric” describes how spacetime curves outside this mass [07:26:00]. It predicts that far from the mass, spacetime is flat, but as one approaches, it becomes increasingly curved, attracting objects and causing time to slow down [07:32:00].

However, the solution revealed two “problem spots”:

1. Singularity at r=0: At the center of the mass, the equation breaks down, indicating a singularity [08:22:00].
2. Schwarzschild Radius: At a specific distance, the Schwarzschild radius, another singularity appears [08:45:00]. At this point, the spacetime curvature becomes so steep that the escape velocity reaches the speed of light [08:57:00]. This means that inside the Schwarzschild radius, nothing, not even light, can escape, leading to the concept of a “black hole” [09:10:00].

The Debate on Black Hole Existence

Initially, most scientists doubted black holes could exist, as they would require an immense amount of mass to collapse into a tiny space [09:26:00]. Astronomers were studying the end stages of stars, where the inward pull of gravity is balanced by outward radiation pressure from nuclear fusion [09:40:00]. When fuel runs out, radiation pressure drops, and gravity pulls stellar material inward [09:52:00].

• White Dwarfs: In 1926, Ralph Fowler proposed “electron degeneracy pressure” as a mechanism to prevent complete collapse [10:01:00]. Due to Pauli’s exclusion principle (electrons cannot occupy the same state) and Heisenberg’s uncertainty principle (constraining position increases momentum uncertainty), electrons exert an outward pressure when compressed [10:10:00]. This leads to the formation of white dwarfs, very dense stars [10:44:00].
• Chandrasekhar Limit: However, in 1930, Subrahmanyan Chandrasekhar realized that electron degeneracy pressure has limits, as electrons can only wiggle up to the speed of light [11:17:00]. This implies a “Chandrasekhar limit” for the maximum mass a white dwarf can support [11:27:00].
• Neutron Stars: Later, it was discovered that stars heavier than the Chandrasekhar limit could collapse further, fusing electrons and protons into neutrons [11:51:00]. Neutron degeneracy pressure, stronger due to neutrons’ greater mass, supports neutron stars [12:05:00].
• Oppenheimer’s Breakthrough: Despite this, scientists held a conviction that something would prevent a star from collapsing into a single point [12:16:00]. This belief was challenged in the late 1930s when Jay Robert Oppenheimer and George Volkoff found that neutron stars also have a maximum mass [12:34:00]. Shortly after, Oppenheimer and Hartland Snyder showed that for the heaviest stars, there is “nothing left to save them” when their fuel runs out, and their contraction “will continue indefinitely” [12:44:00].

Einstein still found it hard to believe, as his math suggested time freezes on the horizon, implying nothing could ever enter a black hole [12:58:00]. Oppenheimer offered a key insight: while an outside observer would never see anything enter a black hole (due to light redshifting and fading) [17:36:00], an observer falling across the event horizon would not notice anything unusual and would pass right through [13:21:00].

The Disappearing Horizon Singularity

The apparent singularity at the event horizon in the Schwarzschild solution (r=2M) was a result of a “poor choice of coordinate system” [16:05:00]. By choosing a different coordinate system, this singularity disappears, confirming that objects can indeed cross into the black hole [16:15:00].

One way to visualize this is by describing space as flowing towards the black hole like a waterfall [16:41:00]. Closer to the black hole, space flows faster. Photons emitted by an infalling object struggle against this flow, becoming dimmer and redder until they fade from view as they approach the event horizon [00:51:00]. At the event horizon, space falls as fast as light can travel, meaning photons get stuck at this infinitely thin boundary [17:09:00]. Inside the horizon, space falls faster than the speed of light, ensuring everything falls into the singularity [17:29:00].

Maximally Extended Solutions and Penrose Diagrams

Different spacetime projections (like different maps of Earth) can reveal different properties [15:53:00]. The Kruskal-Szekeres diagram transforms the Schwarzschild solution so that incoming and outgoing light rays always travel at 45 degrees [20:22:00]. This diagram shows that the black hole singularity is not a place in space, but a moment in time—the very last moment for anything entering the black hole [20:40:00].

The Penrose diagram is an even more compressed map, allowing the entire infinite universe (past, distance, future) to be represented on a single diagram [21:10:00]. In a Penrose diagram for a Schwarzschild black hole:

• Light rays still travel at 45 degrees [21:28:00].
• The singularity is a straight horizontal line at the top, representing a final moment in time [21:42:00].
• The diagram reveals several regions:
  • White Hole: By drawing the past light cone from a point on the event horizon, a new region is uncovered where signals can be sent to the universe, but nothing can ever enter it [22:48:00]. This is the time-reverse of a black hole, expelling matter and light [23:36:00].
  • Parallel Universes: When the diagram is “maximally extended,” it suggests the existence of whole new universes parallel to our own [24:49:00]. This arises from the mathematical possibility of extending coordinates beyond the initial solution [26:36:00].
  • Einstein-Rosen Bridge: The intersection point on the Penrose diagram represents an Einstein-Rosen Bridge, a type of wormhole that could theoretically connect two universes [27:13:00]. However, these wormholes are unstable and pinch off too quickly for anything to traverse them [28:08:00].

Kerr Solution: Rotating Black Holes

The Schwarzschild solution describes a non-rotating black hole, but all stars rotate, and thus all black holes must also rotate to conserve angular momentum [28:57:00]. Solving Einstein’s equations for a spinning mass was much harder, taking 40 years until Roy Kerr found the solution in 1963 [29:12:00].

The Kerr solution presents several dramatic changes:

• Layered Structure: The black hole consists of multiple layers [29:40:00].
• Non-Spherical Symmetry: Rotation causes the black hole to bulge at the equator, making it symmetric only about its axis of spin [29:47:00].
• Ergosphere: Space itself is dragged around with the rotating black hole [30:07:00]. As one gets closer, space is dragged faster, even exceeding the speed of light in the ergosphere [30:13:00]. While here, it’s impossible to stay still relative to distant stars, but escape from the black hole is still possible [30:26:00].
• Outer Horizon: This is the point of no return, where one can only go inward [30:39:00].
• Inner Event Horizon: Inside the outer horizon, a new region allows free movement, meaning one is not immediately doomed to the singularity [30:52:00].
• Ring Singularity: In a rotating black hole, the singularity expands from a point to a ring, which might be traversable [31:12:00].

The Penrose diagram for a spinning black hole shows the singularity lifting and moving to the sides, revealing the new region inside the inner horizon where free movement is possible [31:30:00]. Venturing further could lead to a white hole ejecting one into another universe [31:55:00]. Traversing the ring singularity could lead to an “anti-verse” where gravity pushes instead of pulls [32:30:00]. These maximal extensions suggest an infinite number of universes connected by black holes and white holes [33:14:00].

Limitations and Reality

Despite these fascinating possibilities, the maximally extended Schwarzschild and Kerr solutions are for eternal black holes in an empty universe, lacking a formation mechanism [33:43:00]. This is a primary reason why black holes are realized in our universe but white holes are not thought to exist [34:07:00].

For the maximally extended Kerr solution, an “infinite flux of energy” can pile up along the inner horizon, creating its own singularity and effectively sealing off the ring singularity and any beyond [34:24:00]. This suggests that the exotic features like white holes, wormholes, and other universes predicted by these solutions likely disappear in realistic scenarios [34:50:00].

Traversable Wormholes

In 1987, Michael Morris and Kip Thorne explored the concept of traversable wormholes for interstellar travel [35:10:00]. These theoretical wormholes would have no horizons, be stable, and be constructible [35:15:00]. While some geometries are allowed by Einstein’s general relativity, they require an exotic kind of matter with a negative energy density to prevent collapse [35:39:00]. This type of matter is considered highly improbable under the known laws of physics, making such “science fiction” wormholes unlikely [35:48:00].

According to our current best understanding, it appears that white holes, traversable wormholes, and parallel universes predicted by the maximally extended solutions do not exist [36:22:00]. However, just as black holes were once thought to be impossible, future discoveries and predictions related to general relativity might bring new surprises [36:32:00].