In either classic steady-state or eternal inflation case, energy conservation is not necessarily a problem: you can have vacuum energy that converts steadily into radiation, while being generated by the expansion.
When you have a space that isn't just normal globally euclidean space (such as: on the surface of a sphere), the idea of a vector as in like "a direction you can go in and an amount of how much or how fast or whatever" isn't something that makes sense as something independent of a base location. Instead, each point in the space has associated with it a space of "tangent vectors" at that point, and these spaces are related to each other.
The metric tensor associates to each point a "bilinear form" with some properties, essentially a way of doing something like a dot product of two tangent vectors at that same point.
This in turn allows for defining the notion of the length of some curve through the manifold.
The future, literally. To grossly oversimplify: if all of space is east-west, and time is north-south, the Big Bang is the north pole. Only the universe is the map rather than the globe, and the globe doesn’t have to exist for the map to exist and to have the same expansion in space (/longitude) with respect to time (/latitude).
Also there may or may not be a Big Crunch/south pole, this is all just a way to get into the nature of the geometry by way of a convenient frame of reference.
Noether's second theorem works fine in General Relativity (in fact, this was the historic context of her paper). So for any given time-like vector field, you'll get an energy conservation law. In case of Friedmann cosmology and chosing cosmological time as said vector field, you'll get a term proportional to H² which picks up the change in energy.
However, you won't be able to make this into a covariant expression: Gravitational energy-momentum can be expressed in terms of pseudo-tensors at best...
> Noether's first theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Why is this not obvious? Are not symmetries necessarily the transformations which conserve quantities, and the types of symmetry equivalent to the type of conservation?
Because Noether got a lot of respect for her work despite working (in a time of great sexism, no less!), I know that either I have totally misunderstood or this is a Columbus’ Egg [0] — but I am curious which, and if it isn’t a Columbus’ Egg, what I’ve misunderstood (assuming that sort of thing can even fit into a HN-sized reply and I’m not asking something that would normally be the conclusion of a final year degree level physics module).
A priori, a symmetry is a transformation that, when applied to any valid trajectory, yields another valid trajectory. It is not obvious to me why (certain types of) symmetries necessarily yield conserved quantities...
If I have a motion vector and a force vector, and if work done is ∫ force • displacement dx, and my transformation is one which conserves the scale of and relationship between both vectors (e.g. displacement), then is it not automatically true that work done is also conserved under that transformation?
To get a feel for it (and why it may not necessarily be intuitive) consider the following pairs of symmetries and conservation laws:
- The laws of physics are invariant under time translation (i.e. repeating an experiment at different times gives you the same result ceteris paribus). The corresponding conserved quantity is energy.
- The laws of physics are invariant under spatial translation (i.e. repeating an experiment at two different places gives you the same result ceteris paribus). The corresponding conserved quantity is momentum (this is a vector: one component of momentum for each possible direction of translation).
- The laws of physics are invariant under rotation (i.e. repeating an experiment under different orientations gives you the same result ceteris paribus). The corresponding conserved quantity is angular momentum (again a vector, since the rotation group SO(3) has three generators).
- Electromagnetism, constructed as a classical field, exhibits an internal symmetry in the field quantities. The corresponding conserved quantity is the electrostatic charge (actually a 4-vector current).
It is not a priori obvious from from the kind of arguments you've supplied why these should be the case. Demonstrating these would require access to the Lagrangian formalism (i.e. action principles) and how it behaves under these symmetry operations. I would say that, as you suspect, you are fundamentally misunderstanding the situation.
I’ve heard of the Lagrangian before, but my knowledge of physics is probably somewhere around the level of someone who has only just finished the first year of their degree, at least judging by the modules my alma mater lists for their physics degree (I got Software Engineering from them 14 years ago, all my physics knowledge since then is personal geekery).
What I feel is the core of the resolution of the paradox is (global or local) conservation of energy. Even in an infinitely large eternal steady state universe, if we assume the total energy of the universe is conserved one cannot have an increase in energy density everywhere at once.
If in this universe stars live forever, you'd have eternal "sources" of energy, and for energy to be conserved you'd need compensatory "sinks" draining energy out of the universe, like black holes which don't increase in size. In which the resolution of the Olbers' paradox would be that most of your lines of sight would end in such a black hole.
If, like is usual in physics, you assume local conservation of energy, stars cannot live forever, so in an eternal steady state universe there must be a mechanism recycling the radiation back into a star. In this case, again every line of your sight would eventually hit a star, but most of the radiation would never reach you, being used underway to make a new star. (This is a blatant violation of the second law of thermodynamics of course, which is the actually issue with eternal steady state universes).