Group theory and Spacetime symmetries

In summary, a manifold is considered to be spherically symmetric if its isometry group contains a subgroup isomorphic to SO(3) and its orbits are 2-spheres. This definition is sufficient to calculate the form of a spherically symmetric metric and is related to the space-time symmetries and the familiar rotation group. The symmetries of a manifold are encoded in its set of Killing vectors, and a subset of these can encode information about a smaller symmetry, such as rotational invariance. By using the Lie bracket operation on pairs of vector fields, one can intuitively understand the relationship between the Lie algebra of the Killing vector fields and the Lie algebra of SO(3).
  • #1
cesiumfrog
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"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why?

The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition with no explanation.. except to note that this definition is sufficient to calculate the general form of a spherically symmetric metric/line-element (rather than just writing one down intuitively), which then might be used (in combined with the vacuum-EFE) to obtain Schwarzschild's solution. Could anyone point me to an explanation or proof for the statement?

It does make sense that there should be some relationship between the space-time symmetries and the familiar rotation group, but why do we know the relationship to have this specific form?
 
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  • #2
cesiumfrog said:
"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why?

Think about the more familiar definition of spherical symmetry that one finds in, say, Wald:

A spacetime is spherically symmetric if its isometry group contains a subgroup isomorphic to SO(3) and whose orbits are 2-spheres.

(At least I think that's how Wald defines it; it's certainly the first definition that trips off my tongue, and I know that I learned this kind of thing from Wald.) Can you see how to relate the statement you qouted to the one I've given here?

Presumably, for example, you're aware that the symmetries of a (pseudo-)Riemannian manifold are encoded in the set of Killing vectors (if it exists) of that manifold. And presumably you know that if a complete set of Killing vectors encodes all the information about the symmetries of the manifold, then a subset of the set of Killing vectors could encode information about a smaller symmetry such as the rotational invariance of the manifold?

Can you see a way to combine this fact with the fact that a natural operation on pairs of vector field is the Lie bracket? And can you see how this will lead you to -- at the very least -- an intuitive acceptance of the statement?
 
  • #3


Group theory and spacetime symmetries are closely related concepts in physics, particularly in the context of general relativity. In order to understand the statement provided, it is important to have a basic understanding of these concepts.

Group theory is a mathematical framework that deals with the study of symmetries, transformations, and structures of groups. In physics, it is used to describe the symmetries of physical systems, such as the laws of nature and the properties of particles.

On the other hand, spacetime symmetries refer to the symmetries of the geometry of spacetime. In general relativity, spacetime is described as a manifold with a metric tensor, which represents the curvature of spacetime. Spacetime symmetries refer to the transformations that preserve the metric tensor and therefore, the geometry of spacetime.

Now, going back to the statement, it states that a manifold with a metric tensor is spherically symmetric if the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3). This means that the manifold has a symmetry group that is isomorphic to the rotation group in three dimensions.

This statement is true because the rotation group in three dimensions, SO(3), is the symmetry group of a sphere, which is the most basic example of a spherically symmetric object. This group consists of all possible rotations that can be performed on a sphere without changing its shape or size.

In general relativity, a spherically symmetric spacetime is one in which the geometry is invariant under rotations around a fixed point. This means that the metric tensor remains the same after performing a rotation, indicating a symmetry in the geometry.

Therefore, the statement is saying that if a manifold has a symmetry group that is isomorphic to the rotation group in three dimensions, then it is spherically symmetric. This is because the Lie algebra of SO(3) contains the generators of rotations around a fixed point, which are the Killing vector fields that preserve the geometry of a spherically symmetric spacetime.

In conclusion, the statement is a direct result of the relationship between group theory and spacetime symmetries. It shows that the symmetry group of a spherically symmetric spacetime is isomorphic to the rotation group in three dimensions, which explains the specific form of the Lie algebra of its Killing vector fields.
 

1. What is group theory and how is it related to spacetime symmetries?

Group theory is a branch of mathematics that deals with the study of symmetry and its properties. It is used to describe and analyze the behavior of systems that exhibit symmetries. Spacetime symmetries refer to the symmetries that exist in the fabric of spacetime, which is the four-dimensional continuum in which all physical events take place. Group theory is used to describe and understand these symmetries in the context of spacetime.

2. What are the applications of group theory in physics?

Group theory has many applications in physics, especially in the fields of quantum mechanics and relativity. It is used to describe the symmetries of physical systems, which helps in understanding their behavior and predicting their properties. Group theory is also used in particle physics to classify particles and their interactions, and in crystallography to study the symmetries of crystals.

3. How does group theory relate to the concept of conservation laws?

Conservation laws in physics state that certain quantities, such as energy and momentum, remain constant in a closed system. Group theory provides a mathematical framework for understanding and describing these conservation laws. In fact, Noether's theorem, which relates symmetries to conservation laws, is based on group theory.

4. What are some important groups in group theory that are relevant to spacetime symmetries?

Some important groups in group theory that are relevant to spacetime symmetries include the Lorentz group, which describes the symmetries of special relativity, and the Poincaré group, which combines the Lorentz group with translations in space and time. Other groups that are relevant to spacetime symmetries include the Galilei group, which describes the symmetries of classical mechanics, and the conformal group, which describes the symmetries of conformal field theories.

5. How does group theory contribute to the development of theories in physics?

Group theory plays a crucial role in the development of theories in physics. It helps in understanding the symmetries of physical systems, which in turn leads to the discovery of new laws and principles. Group theory also provides a framework for unifying different theories and predicting new phenomena. For example, the Standard Model of particle physics is based on the symmetries of the underlying gauge groups.

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