How can I repair a crack in my fireplace hearth?
I am not a mason, or an expert on fireplaces. (It is not a bad idea to bring in several such people to get an estimate on the repair.) However, a few things seem fairly clear.

The crack appeared because...


The hearth may have been insufficiently reinforced, thus too little rebar, metal mesh, or nothing at all.
Heat stresses, caused by heating and cooling cycles can cause cracks. A properly built hearth should have been designed to prevent that.
Insufficient foundation under the hearth, which might have allowed it to settle, thus introducing stresses.


One or more of the above factors will have caused the problem. Someone experienced in seeing these types of problems will know what caused it, and thus what is needed to repair the problem so it does not happen again.

For example, it should be obvious if there is no reinforcement at all, since the crack will let you see inside. If there was settling, then one of the pieces may now be tilted relative to the other. There may also be other cracks in the walls or foundation of the house to be found.

If you do nothing, then there are some issues to consider beyond pure aesthetics. The one that comes to mind immediately is the crack will allow infiltration of air into the house from the ground. If there is radon in that air, it could potentially cause lung cancer over a long period of time. While this is not a high probability event, it is possible, and it does happen.

Repair may be a moderately difficult thing, since you really need to eliminate the underlying problem. Simply slapping some mortar on the crack will not solve a thing, even if you bother to find some heat resistant mortar. The crack will just reappear.

Again, I'd suggest contacting a good mason who understands fireplaces and foundations.