Gravitational field has no curl? What about gas discs around stars, black holes, etc.?

When we say that a gravitational field has no curl, we are referring to a fundamental property of gravitational fields in classical physics. This concept is rooted in the mathematical framework of vector calculus, specifically in the context of conservative fields. Let’s break this down and also address your question about gas discs around stars and black holes.

Understanding Gravitational Fields

A gravitational field is a vector field that describes the gravitational force experienced by a mass at any point in space. Mathematically, it can be represented as the gradient of a scalar potential function, known as the gravitational potential. This relationship is expressed as:

• Gravitational Field (g) = -∇Φ

Here, ∇Φ denotes the gradient of the gravitational potential Φ. A key characteristic of conservative fields, like gravitational fields, is that they have no curl. In mathematical terms, this is expressed as:

• ∇ × g = 0

What Does "No Curl" Mean?

The term "curl" refers to a measure of the rotation of a vector field. If a field has no curl, it means that there is no local rotation or swirling motion at any point in that field. For gravitational fields, this indicates that the forces act in a straight line towards the mass creating the field, rather than in a circular or swirling manner.

Gas Discs Around Stars and Black Holes

Now, let’s consider gas discs, such as those found around stars or black holes. These discs are often in a state of rotation and can exhibit complex dynamics. While the gravitational field itself is curl-free, the motion of gas within these discs can create rotational effects.

How Does This Work?

In the vicinity of a massive object, the gravitational field is still conservative and has no curl. However, the gas particles in the disc can have angular momentum, leading to a rotational motion. This is a result of the initial conditions of the gas and the conservation of angular momentum rather than a property of the gravitational field itself.

• Example: Consider a spinning pizza. The dough (gas) rotates around the center (the star or black hole) due to the initial spin imparted to it. The gravitational field from the pizza's center is still curl-free, but the dough is in motion.

Conclusion on Gravitational Fields and Gas Discs

In summary, while gravitational fields are curl-free due to their conservative nature, the dynamics of gas discs around stars and black holes can lead to rotational motion. This motion arises from the initial conditions and the conservation of angular momentum rather than any inherent property of the gravitational field itself. Understanding these distinctions helps clarify how gravitational forces operate in various astrophysical contexts.