Does expansion imply a metric

[Mike2]

Is it possible to say that something is growing and expand, or even proceeding in a causal manner, without a metric? Or does expansion require a metric so that it is possible to compare one state of something to its state from a different time?

[Stevo]

I think so.

We can talk about size without referring to geometry: we can talk about the cardinality of a set. Of course, I think that has the problem then of how to give a "nice" description of local behaviour. Somehow we need to describe the propagation of the elements of the set in terms of causal laws.

I think quantum causal set theory is related to this problem. Not sure though.

[HallsofIvy]

Strictly speaking, even "counting" the cardinality of a set is a (discrete) metric.

In order to be able to talk about "growing and expanding" you need to be able to measure and that is what a metric is.

I don't know what you mean by "proceeding in a causal way"!

[Mike2]

Strictly speaking, even "counting" the cardinality of a set is a (discrete) metric.

In order to be able to talk about "growing and expanding" you need to be able to measure and that is what a metric is.

I don't know what you mean by "proceeding in a causal way"!
I take it that causality requires time to be continuous so that at "every" step of a sequence of events, no matter how small those steps are, you can say that the past is "connected" to the future. It seems that if time is not continuous or connected, then it is not possible to say that two event in ANY way depend on each other. The causal relation is broken if the time betwen them is not connected. Same comment for space.

I just am not sure whether an "amount" of spacetime necessarily required a metric. But if it does then causality proves the existence of a continuous spacetime metric. Next, of course, would be to prove the necessity of the equation for the metric.

[turin]

Causality does not require continuity. Continuity is speculation anyway.

[Mike2]

Causality does not require continuity. Continuity is speculation anyway.
OK, continuity may be too strong a word. I think I only meant connected in the topological sense. I'm wondering if we can derive the metric from these kinds of necessities.