Homogeneity and isotropy

In summary, the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy in a cosmology. However, dropping either of those assumptions could lead to violations of those laws, making it difficult to define and measure these quantities on a global scale. This is due to the lack of time translation symmetry in the FRW metric, which means there is no global notion of the mass/energy of the universe. In general, energy conservation in general relativity depends on how one defines "energy" and "conserved". The differential form extends easily, but the integral form breaks down in curved spacetimes.
  • #1
geoffc
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If the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy, would a cosmology that drops either of those assumptions lead to violations of those conservation laws which in turn could be observed? My first guess would be yes, but from what little I understand it becomes very difficult if not impossible to define and measure those quantities on a global scale. Thoughts?
 
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  • #2
geoffc said:
If the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy, would a cosmology that drops either of those assumptions lead to violations of those conservation laws which in turn could be observed? My first guess would be yes, but from what little I understand it becomes very difficult if not impossible to define and measure those quantities on a global scale. Thoughts?

Sort of. One still has local conservation laws of linear momentum and energy in a small section of empty space, but if the overall cosmology lacks (for instance) time translation symmetry, there won't necessarily be a globally conserved energy.

Note that in spite of homogeneity and isotropy, the FRW metric does NOT have time translation symmetry, which implies there isn't any global notion of the mass/energy of the universe.

See for instance the sci.physics.faq Is energy conserved in General Relativity?, which I will quote in part. For copyright and other reasons I am only quoting a "fair use" portion of the FAQ, so I encourage the OP and other interested people to read the entire original.

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a finite-sized piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

You might also look at the wikipedia article for a more advanced discussion about the various sorts of mass, energy, and momentum defined in GR.
 
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  • #3


Your intuition is correct. The assumptions of homogeneity and isotropy are crucial in the formulation of the conservation laws of linear momentum, angular momentum, and energy. If either of these assumptions were dropped, it would lead to violations of these conservation laws, which could potentially be observed.

To understand why this is the case, let's first define what homogeneity and isotropy mean in the context of cosmology. Homogeneity refers to the idea that the universe looks the same at every point in space, meaning that the distribution of matter and energy is uniform. Isotropy, on the other hand, refers to the idea that the universe looks the same in every direction, meaning that there is no preferred direction in space.

These assumptions are necessary for the conservation laws to hold because they allow us to define and measure these quantities on a global scale. For example, in a homogeneous and isotropic universe, the total linear momentum of the universe must be conserved because there is no preferred direction in which it could change. Similarly, the total angular momentum of the universe must also be conserved because there is no preferred axis of rotation.

If we were to drop either of these assumptions, it would become much more difficult, if not impossible, to define and measure these quantities on a global scale. This would lead to violations of the conservation laws, which could potentially be observed in the form of unexpected changes in the distribution of matter and energy.

In summary, the assumptions of homogeneity and isotropy are crucial in the formulation of the conservation laws in cosmology. Dropping these assumptions would lead to violations of these laws, which could potentially be observed. However, it is important to note that our current understanding of the universe is based on these assumptions, and any new cosmological model that drops them would need to provide strong evidence and explanations for why they are no longer valid.
 

What is homogeneity in the context of science?

Homogeneity refers to the uniformity or consistency of a system or substance. In science, it specifically refers to the uniform distribution of properties or characteristics throughout a system or substance.

What is isotropy and how is it related to homogeneity?

Isotropy is the property of being uniform in all directions. In the context of science, it is closely related to homogeneity as a system that is homogeneous is also isotropic.

Why is homogeneity and isotropy important in scientific research?

Homogeneity and isotropy are important concepts in scientific research because they allow for the simplification of complex systems. By assuming that a system is homogeneous and isotropic, scientists can make accurate predictions and draw conclusions about the behavior of the system without having to account for variations.

What are some examples of systems that are homogeneous and isotropic?

Air, water, and salt solutions are all examples of systems that are homogeneous and isotropic. This means that the properties of these substances, such as temperature or density, are consistent throughout and are the same in all directions.

How do scientists determine if a system is homogeneous and isotropic?

Scientists use various methods and techniques, such as statistical analysis and measurement tools, to determine the homogeneity and isotropy of a system. They may also conduct experiments and observe the behavior of the system to confirm their findings.

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