I have come across the statement that propagation of singularities is a general feature of the hyperbolic equations (context: Einstein Field Equations). Even on an intuitive level, I cannot make sense out of it. To me it seems, that then any initial data we take will already contain the (Big Bang-) singularity, and therefore there is no such thing as a regular initial data.
Since Einstein Equations are hyperbolic, it means then that they will always (?!?) propagate the initial singularity? I would be grateful if anyone could explain how, in the context of general relativity, to understand the statement about the singularity propagation correctly. Thanks!
You're right that singularities generically persist in hyperbolic PDEs. However, we typically don't actually include the instant of the Big Bang in our calculation, since nobody seriously believes GR is the correct theory to use in this regime (we need a full quantum theory at least) so we don't worry about this point.
Instead, we imagine starting a little bit after the initial 'moment', where everything is e.g. a nice smooth FLRW spacetime. You can then follow this back to a singularity, but you don't reach it before your theory breaks down.