[FOM] Could spacetime be discrete?

[Ben Crowell]

Alasdair Urquhart wrote:
>Richard Haney raises the question as to whether spacetime
>could be discrete.  I am not sure what this means, [...]

Smolen's Three Roads to Quantum Gravity explains it well.
It means that spacetime events would be like a network
of discrete points, with each one connected causally to 
some of its neighbors. The causal relationships would not
be fixed, but would be determined in some way by the events
themselves; this is how causality works in classical general
relativity, since, e.g., the formation of a black hole changes
the structure of causality of the surrounding space.

>[...] but if
>it means that there is a minimum length,[...]
 
I'm not an expert, but my general recollection from Smolen's 
book is that notions like length would have meaning only as
large-scale descriptions of the gross structure of the network of
events. Independent of this particular discretized theory of
spacetime, there is something called the Planck length, which
is the only natural scale in a theory of quantum gravity.
Many people have imagined that spacetime would become a sort
of quantum foam at the Planck scale. In string theory, the Planck
scale is the length scale of the strings, and also of the
hypothesized extra dimensions that are curled up.

>[...] this appears inconsistent with special relativity.  Could
>somebody elucidate this point?

It's inconsistent with special relativity, but that's not
surprising, because it's an attempt to imagine a theory of
quantum gravity, and the whole problem with quantum gravity
is that quantum mechanics is inconsistent with general relativity.