Is there not a spacial element in the mix too?
Correct.
There are absolute minimum protected ranges, within which TAs and RAs are triggered, regardless of the rate of closure.
As an example: Without this, it would be possible for an intruder to slowly merge with you, and then by turning towards you generate an increased rate of closure, triggering a rapid succession through TA to RA but without sufficient time for the RA to be effective.
You can only see traffic on the Nav display that TCAS decides may become, or has become, a potential threat.
Sort of. TCAS only displays traffic that it is tracking full time. The extent to which it tracks another aircraft is a mixture of the potential threat posed by the other aircraft AND the congestion on the SSR frequencies.
In busy environments TCAS becomes more 'picky' about which intruders it tracks continuously, and drops those that are not a potential threat. They'll be dropped from the traffic display and TCAS will 'monitor' them at a reduced interrogation rate, upgrading them back to full time tracking if the situation warrants it.
So, in low traffic densities you may see blips out to relatively long ranges (40 miles or so) that are basically zero threat. As you fly into a terminal area such blips will likely be removed from the display, along with possibly aircraft that are closer but with low closure rates, or even quite close but diverging.
As pilots, what would you want in traffic info?
Depends on the service you are giving me and the service being received by the other aircraft. If we are both under radar control there isn't really that much need to tell us about one another, although it might be nice to be told about him if he is constraining our path; that way we won't hassle you for further climb / direct routing.
I wouldn't recommend any assumption on your part that we can see a particular aircraft on our traffic display. There is no gauruntee we have a blip for it, and also no gauruntee that the blip we have is the aircraft you are talking about.
Basically, the Traffic Display has so many limitations it should not be relied upon for anything. Over use (and over optimism about traffic display capabilities!) is a major concern in TCAS operations.
Hope thats of some use.
pb