Null geodesic in 2 dimensional manifold

• paweld
In summary, a null geodesic is a path or curve in a 2 dimensional manifold with a zero tangent vector at every point, meaning it has no length and is essentially a straight line. It is different from a regular geodesic which can have a nonzero tangent vector and a positive length. Null geodesics are significant in understanding the geometry and topology of 2 dimensional manifolds and are used in theories of general relativity and cosmology. They can be calculated using the geodesic equation and can exist in higher dimensional manifolds, although the calculations become more complex.
paweld
I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).

For a perfect donut sitting flat on a table, the circle of contact is a curve on the torus. I may be wrong, but isn't that such a curve without being a null geodesic?

I will now eat the perfect donut.

Only if the metric tensor is nonsingular everywhere.

1. What is a null geodesic?

A null geodesic is a path or curve in a 2 dimensional manifold that has a tangent vector with a zero magnitude at every point along the curve. This means that the path has no length and is essentially a straight line.

2. How is a null geodesic different from a regular geodesic?

A regular geodesic is a path on a manifold that follows the shortest distance between two points. It is not constrained to have a zero tangent vector and can have a length greater than zero. A null geodesic, on the other hand, has a zero tangent vector and therefore has no length.

3. What is the significance of null geodesics in 2 dimensional manifolds?

Null geodesics play a crucial role in understanding the geometry and topology of 2 dimensional manifolds. They are closely related to the concept of light rays in physics and can help determine the curvature and shape of the manifold. They are also important in theories of general relativity and cosmology.

4. How are null geodesics calculated or determined?

Null geodesics can be calculated using the geodesic equation, which is a set of differential equations that describes the path of a geodesic on a manifold. In specific cases, such as in flat Euclidean space, the calculation can be simplified. In more complex cases, numerical methods may be used.

5. Can null geodesics exist in higher dimensional manifolds?

Yes, null geodesics can exist in higher dimensional manifolds. In fact, they are a fundamental concept in the study of higher dimensional geometry and can be used to characterize the properties of these manifolds. However, the equations and calculations may become more complex as the dimension of the manifold increases.

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