Symmetries of Higher Dimensions & Extra Spacetimes

[latentcorpse]

Is it true to say that most higher dimensional spacetimes have symmetries amongst their n extra dimensions?

Thanks.

 

[fzero]

Is it true to say that most higher dimensional spacetimes have symmetries amongst their n extra dimensions?

Thanks.

Most manifolds have no isometries at all, so no. If you are talking about some Kaluza-Klein or string theory model that would try to reproduce standard model physics, then there are additional constraints on the allowed manifolds that require certain symmetries.

 

[latentcorpse]

Most manifolds have no isometries at all, so no. If you are talking about some Kaluza-Klein or string theory model that would try to reproduce standard model physics, then there are additional constraints on the allowed manifolds that require certain symmetries.

Yeah. Thanks.

Just to clear up what an isometry actually is though:

I know it is a symmetry transformation of the metric tensor field.
i.e. a map $\phi: M \rightarrow M$ such that $(\phi)_*g=g$

However, what does this actually mean?

Suppose the metric is invariant of t. We can see from Killing's equation that $\frac{\partial}{\partial t}$ will be a Killing vector field easily enough but what does this actually mean the isometry is?
Would the isometry be the map $\phi: M \rightarrow M ; t \mapsto t+c$ i.e. time translations?

Thanks.

 

[fzero]

Yeah. Thanks.

Just to clear up what an isometry actually is though:

I know it is a symmetry transformation of the metric tensor field.
i.e. a map $\phi: M \rightarrow M$ such that $(\phi)_*g=g$

However, what does this actually mean?

Suppose the metric is invariant of t. We can see from Killing's equation that $\frac{\partial}{\partial t}$ will be a Killing vector field easily enough but what does this actually mean the isometry is?
Would the isometry be the map $\phi: M \rightarrow M ; t \mapsto t+c$ i.e. time translations?

Thanks.

Yes, any isometry corresponds to the existence of a Killing vector field. From the Killing vector, we can define a conjugate coordinate, at least locally, for which the isometry corresponds to a translation.

 

[paranormal]

I can say that while it is true that many higher dimensional spacetimes have symmetries among their extra dimensions, it is not necessarily a universal truth. The existence of symmetries in higher dimensions is dependent on the specific model or theory being studied.

Some theories, such as string theory, propose the existence of extra dimensions that are compactified and have symmetries among them. This is because these symmetries play a crucial role in the mathematical framework of string theory, known as supersymmetry.

However, other theories, such as brane cosmology, do not require the existence of extra dimensions with symmetries. In fact, the brane world model proposes that our observable universe is a 3-dimensional brane embedded in a higher dimensional spacetime, where the extra dimensions are not necessarily symmetric.

Therefore, it is important to note that while symmetries among extra dimensions may be present in some models, it is not a universal feature of all higher dimensional spacetimes. Further research and experimentation are needed to fully understand the nature of these extra dimensions and their potential symmetries.