I have a simple cosmological riddle, and the answer, which seems to imply the existence of an absolute frame of reference for every model universe of finite age that has a Lorentzian spacetime structure: “Why is the universe not infinitely old? Because waiting an infinite amount of time takes too much time, especially toward the end.” http://everythingimportant.org/relativity/special.pdf How could a Lorentzian spacetime structure of finite age not have an absolute frame of reference?
In other words, you don't know how to create a mathematical model of a Lorentzian spacetime structure of finite age (finite history) that has no absolute frame of reference.
Alexander Friedmann proposed a model consistent with GR in 1922. http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric Here's a solution for a flat universe dominated by dust and a cosmological constant and isotropic components of the metric \(T\) in Einstein's equations. \(-c^2 d\tau^2 = -c^2 dt^2 + a(t)^2 ( dx^2 + dy^2 + dz^2) a(t) = a_0 \left( \frac{t}{t_0} \right)^{\tiny \frac{2}{3} } \frac{d a(t)}{dt} = \frac{2}{3} \frac{a(t)}{t} \frac{d a(t)}{dt} = -\frac{2}{9} \frac{a(t)}{t^2} \rho(t) = \frac{3}{8 \pi G a^2(t) } \left[ \left( \frac{d a(t)}{dt} \right)^2 \right] - \frac{\Lambda c^2}{8 \pi G} = \frac{3}{8 \pi G a^2(t) } \left[ \frac{4}{9} \frac{a^2(t)}{t^2} \right] - \frac{\Lambda c^2}{8 \pi G} = \frac{1}{8 \pi G} \left[ \frac{4}{3 t^2} - \Lambda c^2 \right] p(t) = \frac{\Lambda c^4}{8 \pi G} - \frac{c^2}{8 \pi G a^2(t)} \left[ 2 a(t) \frac{d^2 a(t)}{dt^2} + \left( \frac{d a(t)}{dt} \right)^2 \right] = \frac{\Lambda c^4}{8 \pi G} - \frac{c^2}{8 \pi G a^2(t)} \left[ -\frac{4}{9} \frac{a^2(t)}{t^2} + \frac{4}{9} \frac{a^2(t)}{t^2} \right] = \frac{c^4}{8 \pi G} \Lambda \)
That doesn't answer my question: How could a Lorentzian spacetime structure of finite age not have an absolute frame of reference?
The "how" is General Relativity -- for I have given you a Lorentzian structure (first line is a metric) and this structure has a locally-defined frame of rest given by its isotropic matter content (this is not a preferred coordinate frame, but rather one special to the matter distribution being discussed) and a well defined notion of hypersurfaces of infinite spatial extent of constant t-parameter that all observers co-moving with the isotropic dust will agree is a time-coordinate that terminates at t=0 at infinite density.
Essentially, an absolute frame of reference means that there is a frame-dependent law of physics that would enable an experimenter to distinguish one frame over another. One could call “co-moving with isotropic dust” to be a frame-dependent law of physics but suppose that your isotropic dust doesn’t interact with ordinary matter. Let’s suppose that we’re in a universe described by a metric determined by this undetectable isotropic dust. Would it still be possible to distinguish one inertial frame over another?
My mistake was omitting the "and" between metric (which I displayed explictly) and T (for which I displayed the uniform density and isotropic pressure terms).
