Demystifying the event horizon: Newton vs Einstein

The event horizon may sound intimidating, yet it is a surprisingly simple concept. Much of its mystery comes from the language used in its description, rather than from the physics itself. To develop an intuitive understanding, we replace Einstein's relativity with Newton's laws. Einstein's viewpoint is necessary for precise calculations. In most other cases, Newtonian physics is sufficient. It provides all the basic elements needed to explain what the event horizon is and how it is formed.

Fig. 1. An artistic impression of a black hole’s event horizon, the boundary beyond which nothing can escape.

For a star to become a black hole, it must be at least 20 times more massive than our Sun. Nevertheless, we can use our planet to illustrate what leads to the formation of an event horizon, as the theoretical model remains the same. Starting with familiar surroundings makes the concept more approachable. Once the fundamentals are clear, transitioning to Einstein’s interpretation becomes far less daunting.

In Newtonian terms, an event horizon can be described as a radius at which the escape velocity equals the speed of light.

Within this radius, gravity is so strong that an object would need to exceed the speed of light to escape. Given that the speed of light is the universal limit, everything that crosses the event horizon gets trapped inside. In this sense, the event horizon outlines a region of no return for all known particles, including photons.

Escape velocity determines what gets trapped and what breaks free

To see how escape velocity works, imagine a comet (the red body in Fig. 2) approaching Earth from deep space. Depending on its speed and direction, the comet faces three possible outcomes: it may collide with Earth (beige path), become gravitationally bound and enter an orbit (green path), or escape Earth’s gravity altogether and continue its journey through space (blue path). The determining factor in all three outcomes is the escape velocity. Escape velocity (brown path) is the minimum speed an object must have at a given distance (r) to overcome the gravitational pull of the massive body (M) and reach infinity without further propulsion.

Fig. 2. Escape velocity, V(escape) and possible trajectories near Earth

Earth’s gravity pulls the comet inward, while the comet’s forward motion carries it sideways. The balance between these two effects determines the comet's trajectory. If the comet’s speed is precisely equal to the escape velocity, it follows a parabolic path that takes it away from Earth, but only just. If its speed is greater than the escape velocity, it follows a hyperbolic path and escapes with energy to spare. However, if its speed is even slightly lower, gravity wins, and the comet becomes bound to Earth, tracing out an elliptical orbit or colliding with the planet. Because of its precise value, the escape velocity marks the boundary between trajectories that remain bound and those that escape into space.

Preparing for the leap to black holes

Escape velocity depends solely on the strength of gravity, which is set by the large body's mass (M) and the distance(r) from that body. This relationship is represented by a simple formula, where Vesc is the escape velocity, G is the universal gravitational constant, M is the mass of the large body, and r is the distance from the large body's center to the small body's center.

Crucially, the formula does not depend on the mass of the escaping object. It applies equally to any object with a mass much smaller than M, whether a comet, a spacecraft, or even a particle with negligible mass and high speed. The relationship between escape velocity, Vesc, and radius, r, shows that as the radius decreases, the required escape velocity increases.

Extending this line of reasoning suggests that at a specific radius r, the escape velocity must exceed the universal limit. This radius can be found by setting Vesc = c, which yields a value that coincides with the Schwarzschild radius rs, introduced in Einstein’s general theory of relativity.

The Schwarzschild radius marks the boundary beyond which parabolic and hyperbolic escape trajectories are no longer available. This boundary forms a spherical surface known as the event horizon. Within this region, gravity becomes so strong that even the fastest particles, photons, get trapped. In the area of the event horizon, all available paths lead inward, while outward trajectories are closed off by the universal limit imposed by the speed of light.

Einstein’s perspective: gravity without force

Einstein transforms the concept of gravity into the geometry of spacetime. In classical physics, Earth generates a gravitational field whose strength determines how much trajectories are bent. The paths of comets and satellites are calculated by measuring this field. General relativity takes a different approach. Instead of curving trajectories, Earth curves spacetime itself. In this picture, comets don't experience a force exerted on them by Earth, but follow straight paths, called geodesics, in the curved spacetime. To calculate trajectories, spacetime curvature is measured rather than the field.

The event horizon: where Newton and Einstein meet

In Newton’s picture, gravity traps objects by force; in Einstein’s picture, spacetime traps them by geometry, but both point to the same boundary: the event horizon. Because the field and the curvature are spherically symmetric around a massive body, both models agree on the value of the Schwarzschild radius. When the escape velocity is set equal to the speed of light, Vesc = c, the resulting radius matches the Schwarzschild radius derived from Einstein’s equations.

The agreement is not accidental. It reflects two different mathematical descriptions of the same underlying phenomenon: one expressed in terms of force, the other in terms of geometry. General relativity doesn't rely on escape velocity to counteract gravity because gravity isn't present in the traditional sense.  In this framework, a massive body like Earth curves spacetime, and a passing object like a comet follows a geodesic, determined by its velocity. So, instead of asking how strong gravity should be to trap light, Einstein’s theory considers how curved spacetime must be to direct all geodesics inward the event horizon.

In general relativity, an event horizon is defined by a radius at which all possible geodesics are directed inward.

Here, the event horizon is a region around a massive body where spacetime is so curved that no outward paths exist.