Does homogeneity infer isotropism?

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Discussion Overview

The discussion centers on the relationship between homogeneity and isotropism in the context of cosmology and general relativity. Participants explore whether the property of homogeneity necessarily implies isotropy, examining theoretical implications and providing examples from physics.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that homogeneity implies isotropy, suggesting that if the universe is uniform everywhere, it should appear the same in all directions.
  • Others present counterexamples, such as an infinite universe with a uniform vector field, where homogeneity exists without isotropy due to a preferred direction.
  • One participant provides specific metrics to illustrate that a homogeneous metric can be isotropic, while an inhomogeneous metric can still be isotropic, challenging the direct implication between the two properties.
  • Another participant asserts that homogeneity does not imply isotropy, using the example of a cylinder that is homogeneous but not isotropic.
  • A question is raised regarding the implications for infinite spacetime and how it relates to the concepts of homogeneity and isotropy.
  • A reference to a text suggests that only observers moving with the cosmological fluid can perceive isotropy, indicating that homogeneity may not lead to isotropy for all observers.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between homogeneity and isotropy, with no consensus reached. Some support the idea that homogeneity implies isotropy, while others provide counterarguments and examples that suggest otherwise.

Contextual Notes

Participants reference specific metrics and theoretical constructs, indicating that the discussion involves complex mathematical and conceptual frameworks. There are also indications of uncertainty regarding the accuracy of some claims, particularly from those who are still learning about general relativity.

Loren Booda
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Does homogeneity infer isotropism?
 
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I believe homogeneity does imply isotropism. If the universe is the same everywhere than it should look the same in every direction. What am I missing?
 
Loren Booda said:
Does homogeneity infer isotropism?

See here and here. Or, just consider an infinite universe with a uniform, non-zero vector field. Every point can have the same value of the field (implying homogeneity), but it will have a preferred direction (implying non-isotropy).

If enforced at all points, however, isotropy does imply homogeneity.
 
I don' think so,

you can have homog /inhomog conditions and isotropic and uniso... cdtns. They do not imply the other. Consider the metric:

ds^2=dt^2-dx^2-dy^2-dz^2

This is homogeneous and isotropic.

Now consider

ds^2=dt^2-a_1(x)dx^2-a_2(y)dy^2-a_3(z)dz^2

This is inhomogeneous but still isotropic,

lastly consider:

ds^2=dt^2-a_1(x,t)dx^2-a_2(y,t)dy^2-a_3(z,t)dz^2

This is inhomog. and unisotropic.

I am just learning GR but I think this is right. The general gist is right - there might be a couple of hiccups in the equations.
Maybe someone with more experience can set me right if I am wrong.
 
Loren Booda said:
Does homogeneity infer isotropism?
No. Homogeneity does not imply isotropy. The cylinder is clearly homogeneous but not isotropic.
 
How about for an infinite spacetime?
 
"Clearly, the universe cannot look isotropic to all obervers...Only an observer who is moving with the cosmological fluid can possibly see things as isotropic." --MTW, p. 714.

If the universe is homogeneous, then there can be one and only one local frame in which the universe looks isotropic. The CMB rest frame is fairly well defined to be approximately 368 km/sec (relative to the Sun) in the direction of the constellation Leo, and this is the local frame in which the cosmic microwave background radiation looks isotropic. Is there any other frame that is a more likely candidate to be our local comoving frame?
 

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