Question

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of these black holes exerts the weakest tidal forces on an object near its event horizon? (a) a \(10 M_{\text {Sun }}\) black hole (b) a \(100 M_{\text {sun }}\) black hole (c) a \(10^{6} M_{\text {Sun }}\) black hole

Short answer

(c) A \(10^6 M_\text{Sun}\) black hole exerts the weakest tidal forces.

Step-by-step solution

Understanding the Problem

The exercise requires us to identify which black hole among the given options exerts the weakest tidal forces on an object near its event horizon. Tidal forces are generally stronger when the distance between the object and the center of the black hole is smaller.

Concept of Event Horizon and Tidal Forces

The event horizon is the boundary surrounding a black hole, beyond which nothing can escape. Tidal forces are differential gravitational forces across an object, and they are strongest where the gravitational gradient is steepest. The smaller the black hole, the closer the event horizon is to the singularity, and thus, the stronger the tidal forces.

Relating Mass to Tidal Forces

The tidal force on an object at the event horizon depends inversely on the square of the black hole's radius but directly on the black hole's mass. This means larger black holes have larger event horizons, reducing the steepness of gravitational gradients and thereby decreasing tidal forces.

Calculating Tidal Forces

For a black hole of mass \(M\), the dominant tidal force at its event horizon scales as \(\frac{1}{r^3} = \frac{1}{(2GM/c^2)^3} \), where \(r\) is the radius of the event horizon. Therefore, as \(M\) increases, \(r\) increases, resulting in weaker tidal forces for larger black holes.

Choosing the Correct Answer

The given options are a \(10 M_{\text{Sun}}\), \(100 M_{\text{Sun}}\), and \(10^6 M_{\text{Sun}}\) black holes. According to our explanation, the largest black hole, \(10^6 M_{\text{Sun}}\), will have the weakest tidal forces at its event horizon due to the larger radius.

Key concepts

Tidal Forces

Tidal forces occur due to gravitational differences across an object when it is near a massive body like a black hole. These are differential forces that can cause stretching or distortion. Consider how the moon's gravity causes tides on Earth. This is a result of tidal forces.

When an object is close to a black hole, the side facing the black hole experiences a stronger gravitational pull than the opposite side. This gradient causes tidal forces.

Black holes are especially known for their strong tidal forces because they have a steep gravitational gradient. The closer you get to the center of the black hole, the stronger the tidal forces become due to the more intense gravitational pull.

Thus, in the context of black holes, tidal forces are very important since they determine how much an object might stretch or even get torn apart as it approaches. This is often dramatized in sci-fi as being 'spaghettified' if the tidal forces become extreme.

Event Horizon

The event horizon of a black hole is like an invisible boundary surrounding it. It's the point at which the gravitational pull becomes so strong that nothing, not even light, can escape. Imagine trying to swim out of a perfectly smooth whirlpool—right at the edge, the force is just too strong to overcome.

Once an object crosses the event horizon, it's as good as gone, forever hidden from the outside universe. This boundary does not have a solid surface; think of it as a point of no return.

The size or radius of the event horizon depends on the mass of the black hole: more massive black holes have larger event horizons due to their stronger gravitational influence. So if you're comparing two black holes, the one with the larger mass generally has a larger event horizon.

Gravitational Gradient

The gravitational gradient is like a measure of how quickly gravity changes when you move through space. In terms of black holes, it's the variation in gravitational force over a distance.

In simpler terms, a steep gravitational gradient means the force of gravity changes rapidly as you get closer to or farther from a black hole. Especially near small black holes, this gradient is quite sharp, causing stronger tidal forces.

Larger black holes, due to their size, spread their gravitational influence over a greater area, resulting in a more gentle gradient. This means that as you get closer to a larger black hole's event horizon, you won't experience as dramatic a change in gravitational force.

Singularity

The singularity is at the very heart of a black hole, where the gravitational forces crush all matter to an infinitesimally small point. It's here that the laws of physics as we know them break down.

At the singularity, density becomes infinite, and isn't truly describable using current physics models. This is why it's often shrouded in mystery and speculation in both scientific and science fiction communities.

In terms of your journey towards a black hole, the closer you get to the singularity, the greater the gravitational pull you face, contributing to increased tidal forces. However, since the singularity is hidden behind the event horizon, direct observation or contact with it is impossible according to our current understanding. This is why understanding other models that could replace conventional physics, particularly quantum gravity, continues to be an intriguing challenge for physicists.