Explanation of homogeneity of space and time by giving examples?

First some definitions:

• Homogeneity means that something looks the same at every point;
• Isotropy means that something looks the same in every direction.

Homogeneity of Space: No point is space is special, so the same basic laws of physics should govern all of space. For instance, if electrons repel each other on Earth, we don't expect electrons to attract each other in the Andromeda Galaxy. More generally, if Maxwells equations hold on Earth, we also assume that they hold in the rest of the universe.

Homogeneity of Time: No point in time is special, so the same basic laws of physics should govern all of time. So again, if Maxwells equations are valid today, there is no reason to expect the equations to suddenly become invalid tomorrow.

Isotropy of Time: No direction in time is special. One way to visualize this, is to look at a simulation of Brownian motion for a classical gas at equilibrium, and then run the video in reverse -- the particles behave in the exact same way!

The other answers are right, but I'll take a different approach.

The three properties you mention reflect what in physics is called symmetry.

A symmetry means that something remains the same under some transformation.

In our case:

Homogeneity of space means that the Physics doesn't change (it's symmetric) under space translations.

Homogeneity of time means that the Physics doesn't change under time translations.

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.

For example, you are measuring the period of a pendulum on the Earth. If you go to the Moon, it changes (you would be able to distinguish where you are). Does this mean that the space is not homogeneous? No, it means that you have to consider the effect of gravity. So you would have to move the Earth-Moon system. Of course, you can't move everything in the Universe because then you a pure tautology.

So when you do some transformation you have to do an extra effort and rearrange some parts of the experiment.

To conclude, the is an important theorem (Noether's theorem) which says:

For every symmetry the is a conserved quantity.

You can prove that homogeneity of space means that momentum is conserved and time homogeneity implies that energy is conserved.

Those statements are VERY abstract results of what is called Noether's Theorem. In plain english, it says that:

1. The laws of physics are the same here as anywhere else (e.g., General Relativity holds at all places in space.)
2. The laws of physics are the same for all possible values of the time coordinate. F=ma now and at 10,000,000 BC for example, and into the future.
3. The laws of physics do not depend on the direction of time. The laws work if you run time forwards or backwards. This is slightly confusing, since time seems to run forward. It is still not fully solved why this is. See arrow of time.