# How can a continuous spacetime look discrete on a larger scale?

Referring to Arnold Neumaiers answer here, where he states that:

"If time appears discrete (or continuous) at some level, it could still be continuous (or discrete) at higher resolution"

This seems to imply that a measurement of a discrete space-time at some resolution does not imply "fundamental" discreteness and hence discrete and continuous space-times are equivalent descriptions.

So I would like to know:

(How) Can a continuous (space-) time look discrete on a LARGER scale?

i. e. how could (space-) time at (1cm) 1s resolution look discrete while being continuous at a (1mm) 1ms scale?

• The answer you quote talks about time at a higher resolution appearing discrete, but your question poses it the other way round - measuring at 1ms intervals is higher resolution than measuring at 1s intervals. – Rob Lambden Aug 25 at 22:32
• As far is I understand it the answer by arnold states it both ways in brackets – Katermickie Aug 25 at 22:35

"If time appears discrete (or continuous) at some level, it could still be continuous (or discrete) at higher resolution"

This seems to imply that a measurement of a discrete space-time at some resolution does not imply "fundamental" discreteness

This is correct - there can be detail which is continuous but which you have not the ability to resolve and hence only see an apparently discrete change.

and hence discrete and continuous space-times are equivalent descriptions.

This is not necessarily the case.

We don't really know the consequences of such discrete space-time, but we would presumably reach a point at which resolving differences between discrete and continuous is possible. Put crudely, you'll notice an edge.

However we would expect a discrete space-time to have a limiting case (when considered at a large enough scale) to match our continuous models, simply because we know those models work well at the appropriate scales.

So I would like to know:

How can a continuous (space-) time look discrete on a LARGER scale?

The argument is fairly simple. Imagine on a very detailed scale (a small scale) a graph has a level region which then develops a very steep (but continuous) slope to a new level.

Now "zoom out". Zooming out the steep slope stops looking like a slope and starts appearing like a vertical line. You cannot, with measuring tool scaled to that level of "zoomed out" detail, resolve the co0ntinuous slope from one level to another and it appears simply as a sudden change from one discrete level to another - a step.

So at a larger scale it can seem discontinuous, but on a more detailed, smaller scale, it can appear continuous.

• I would assume that if a discrete spacetime will look continuous at large scales and a continuous spacetime can look discrete, you would never know if you measured the "fundamental level" when observing either of both. In my understanding this implies equivalence of both. You can always describe one as the limiting case of the other – Katermickie Aug 25 at 22:51
• They might be said to give equivalent results at a specific level of measurement, but that does not mean they are equivalent. The measurement part is very important in physics. You would define the limit of both models with reference to where one starts failing to match the other (or both). You might never be able to analytically generate a non-limiting version of one theory without simplifying it to become the other. – StephenG Aug 25 at 23:17