Black holes induce time dilation primarily because of their strong gravitational fields, rather than the speed required to stay outside their event horizons. Time dilation near a black hole is a consequence of the general theory of relativity, which describes gravity as the curvature of spacetime caused by mass and energy.
According to general relativity, the presence of a massive object like a black hole warps the fabric of spacetime around it. This warping affects the flow of time, causing it to pass at different rates depending on the strength of the gravitational field.
As an observer approaches a black hole or enters its vicinity, they experience a stronger gravitational field. The closer they get to the black hole, the more intense the gravitational pull becomes. This leads to a significant time dilation effect, known as gravitational time dilation.
Gravitational time dilation near a black hole can be described by the following equation:
Δt' = Δt √(1 - (2GM / rc^2))
where Δt' is the time interval experienced by an observer close to the black hole, Δt is the time interval measured by a distant observer (far from the gravitational influence), G is the gravitational constant, M is the mass of the black hole, r is the distance from the center of the black hole, and c is the speed of light.
In this equation, the term (2GM / rc^2) represents the gravitational potential, which is a measure of the strength of the gravitational field. As the gravitational potential increases (as an observer gets closer to the black hole), the square root term approaches zero, leading to a significant time dilation effect. This means that time appears to pass more slowly for an observer close to the black hole compared to an observer further away.
The velocity required to escape the event horizon of a black hole is extremely high, approaching the speed of light. However, the primary factor causing time dilation near a black hole is the gravitational field rather than the velocity. The gravitational time dilation near a black hole would occur even if an observer were stationary with respect to the black hole's center.
In summary, while velocity does affect time dilation in special relativity, the time dilation induced by black holes is primarily due to their strong gravitational fields as described by general relativity.