Search results
Results from the WOW.Com Content Network
Notice that both homogeneity and isotropy are scale-dependent quantities: they depend on the spatial scale where we choose to effectuate our measurements. To give you a specific example, consider steel: steel is an iron-carbon alloy. At a large enough scale (let's say the mm scale), steel is homogeneous.
2. The unit vector r shows that Coulomb's force is parallel to the line joining the charges. It could not be otherwise unless space itself has some built in directional property, for with two point charges alone in empty and isotropic space, no other direction could be singled out. What is an isotropic space?
Isotropy of time (are you sure that it's isotropy of time and not isotropy of space, I don't recall Landau mentioning it) means that the Physics doesn't change if you go backwards in time (but this if false, because the weak interaction violates time reversal). But two experiments apparentely might not obey this properties.
Modern cosmology is built on the Friedmann equations, which in turn rely on isotropy — the idea that the universe looks the same in every direction — as a fundamental assumption. However, there's a very noticeable dipole in the CMB, the standard interpretation of which is that we are moving with respect to the frame in which the CMB is at rest.
In order to link homogeneity and isotropy, we may invoke the "Copernican principle," that we do not live in a special place in the universe. Then it follows that, since the universe appears isotropic around us, it should be isotropic around every point; and a basic theorem of geometry states that isotropy around every point implies homogeneity.
This question comes from exercise 27.1 in Gravitation by Misner, Thorne and Wheeler. They required the following: Use elementary thought experiments to show that isotropy of the universe implies homogeneity. I know homogeneity as the universe is the same everywhere at a given time, and isotropy is related to direction.
If one starts off with the assumption of homogeneity and isotropy, then I can definitely see why the transformations should be linear, since homogeneity requires that the form of transformation should not depend on the location of the two inertial frames in space, this the derivative of the transformation should be independent of location, i.e ...
1. As per my understanding so far, homogeneity of space doesn't require a special vantage point (all points in space are "equivalent" to each other) and is a universal statement in that sense; whereas for the isotropy of space, we talk with respect to a single point (the points on any circle, centered at this point, are all "equivalent" to each ...
In this case isotropy is a characteristic of the emitter. If I would interpret isotropy in terms of noise, I would understand it as being the same in all directions with respect to the receiver, e.g. regardless of direction, the received signal would have identical noise functions that were added to the signal.
1 Answer. Sorted by: 1. I think I have found an answer to my question. The answer is that the first notion is the correct definition and the second notion of isotropy follows from the first notion. This goes as follows: If dσ1 d σ 1 and θ, ϕ θ, ϕ are given, where θ, ϕ θ, ϕ are measured with respect to the normal vector of dσ1 d σ 1 ...