Correlation does not imply causation; but what about when one of the variables is time? I know this question has been asked a billion times, so, after looking online, I am fully convinced that correlation between 2 variables does not imply causation. In one of my stats lectures today, we had a guest lecture from a physicist, on the importance of statistical methods in physics. He said an astounding statement:

correlation does not imply causation, UNLESS one of the variables is time. So, if there is a strong correlation between some independent variable and time, then this implies causation as well.

I had never heard this statement before. Do physicists/relativists see "Causation" differently than stats people?
 A: I have not heard this before, and it would not be true according to the conceptions of causality that I am familiar with (though I am not a physicist).
Typically, for $X$ to cause $Y$ it is necessary that $X$ precedes $Y$ in time. So if $Y$ precedes $X$ then it cannot be "caused" by $X$, regardless of any correlation. Moreover, $X$ preceding $Y$ is not a sufficient condition for causation (also regardless of any correlation).
A: I don't think time is necessarily unique in this, but it's certainly a good example. The point is that typically if A & B are correlated, you can surmise that there is some common causality, but you don't know whether A causes B or B causes A, or perhaps a third variable C causes both A & B. However, in certain cases, you can rule out that any other variable caused A, and so it must be that A caused B. One such example is a controlled experiment, where you, the experimenter, control A. Then if the change you make in A "correlates" with a change in B, you know it must have been A that caused B to change, not the other way around.
Another type of scenario, which is the one this example with time falls into, is if you simply know that no other variable could have caused A because you know that nothing whatsoever can influence A. Since time just flows by one second at a time regardless of any other variable in the world, then if time correlates with changes in some variable your interested in (say, the number of people on the planet), you know for sure that the passage of time must have caused that variable to change, rather than your variable causing time to pass or otherwise change (i.e. time didn't go forward because more people were born, it has to be the other way around).
What you still don't know, of course, is whether the causality is direct. Presumably the passage of time itself doesn't automatically produce more human beings. Rather, history unfolding causes progress in various aspects of society, and this causes the population to increase in size (and even that is a simplification of many little causative relationships). But regardless of the precise factors at play, you do definitely know that A (ultimately) leads to B and not the other way around.
A: Actually, correlation does imply a causal relationship.
Perhaps A caused B, or C caused A and B.
However, correlation does not prove causation.
This is self-evident.
A: I'll provide another answer, since I think the ones currently provided miss an important point of the statement the physicist made. The quoted statement is:

"correlation does not imply causation, UNLESS one of the variables is time. So, if there is a strong correlation between some independent variable and time, then this implies causation as well."

The physicist is not saying:

"If X and Y are correlated, and X comes before Y, then the correlation implies causation."

That would be incorrect. What the physicist is saying is:

"If X and time are correlated, then that correlation implies that increasing time causes an increase (or decrease) in X."

An example might be entropy. If we have a strong correlation between time passing and entropy increasing, then we might say that increasing time causes an increase in entropy. Note that this ignores what the physical causes of the increasing entropy might be (particle decay, expanding universe, etc.).
One of the traditional requirements for causation is time progression, namely that X can only cause Y if X comes before Y. But if one of your variables IS time, then time progression is already built into the relationship (if a relationship exists).
EDIT: Based on a variety of comments, I'm going to add the following. I think that the physicist may be using a different idea of the word "causation" here. He seems to be saying that if there is a correlation between an independent variable and time, you can conclude that the independent variable changes predictably as time passes. Some people might say the changes are "caused" by time passing, this isn't really how statisticians use the words "cause" or "causation", so that may be causing some of the confusion.
A: I would interpret this as a semantic rather than mathematical/statistical argument. I would also take it as a rather severe generalization.
The Bradford Hill Criteria, often used in epidemiology, provide a good framework for thinking about causation. Nothing can definitively prove causation, whether or not time is a factor, and I suspect that the lecturer was not trying to make such a strong assertion. However, many different factors can be used as reasonable arguments for causation.
For example, the Bradford Hill criteria suggest that strength of association between variables can provide evidence for causation, but is not on its own sufficient. Similarly, an association that is consistent with other known/believed facts may suggest causation more strongly than an association that is inconsistent with prevailing knowledge. Temporality is also one of the criteria -- a cause should precede its effect. An association, and the inferences we make about causation, must make temporal sense. I recommend reviewing the other criteria. Some are specific to epidemiology and aren't as applicable to physics but it's still a useful way of thinking.
The main point is that, while no single piece of evidence is going to definitively prove causation, you can build a good case for it based on a number of different logical checks. I would argue that giving absolute precedence to any one criterion, such as time, is not appropriate, but temporality can be an important factor when making a case that causation is plausible.
This leads to a broader point about statistics: generally speaking, we use statistics to make an argument. We use data and statistical tools to make a certain point. Often, the same data (and even the same tools) can be used to make conflicting points. We can't locate the definitive proof of causation in the math itself, but we can deploy our statistical tools as part of a broader argument. For more on that, I recommend Abelson's Statistics as Principled Argument.
To loop this back to the original situation, let's say you've done an experiment about the effect of a the concentration of a certain chemical in a solution on the temperature of that solution. You suspect that adding more of this chemical will result in a reaction that increases temperature. You add more gradually over time. You can look at temperature against time and see an increase. All this shows is that temperature is increasing with time; it doesn't prove that time itself (or anything else, for that matter) has some causal effect. It does, however, provide some evidence in a broader argument that increased concentration of this chemical results in a reaction that increases temperature.
A: The sentence is quite simple and not worth overthinking (and has nothing to do with precedence). 
If there is an established correlation between a variable and time (i.e. we know that an increase in time is accompanied by an increase in the variable, and this is a given), then we know the "causal" direction: i.e. time increasing, causes the variable to increase.
Because the alternative hypothesis of "nah-uh, it could be that time only increased because the variable increased first" simply cannot stand given the way time works.

This might sound like a silly observation, but it has important implications for study design trying to prove a causal direction. An important example in medicine is the difference between doing a cross-sectional and a cohort study.
E.g., a cross-sectional study trying to find a link between smoking and cancer might take a group of people, divide it into smokers vs non-smokers, and see how many in each group have cancer vs no-cancer. However, this is weak evidence because a correlation between smoking and cancer could also be interpreted as "people who have cancer are more likely to enjoy taking up smoking".
However, if you perform a cohort study, i.e. take a group of smokers and a group of non-smokers, and follow them up through time, and measure the variable "cancer in smokers minus cancer in non-smokers", and establish a positive correlation of this variable with time, (under reasonable assumptions, such that smoking amount once started is constant and independent of time etc) then you know that "time" is the cause of the cancer difference, since you cannot claim that increased rates of cancer caused time to pass more in the smoking group. Therefore you can claim a causation between time passing and a positive cancer difference related to higher rates in the smoker group. (or, more simply stated, time spent belonging to the smoking group causes a proportional increase in cancer risk).
Furthermore, the weakness of the cross-sectional study, i.e. the possibility that "people with cancer are more likely to take up smoking" has now gone out the window, since smoking as a variable has been taken out of the "time vs cancer" equation (here assumed to be constant and therefore unaffected by time). In other words, by formulating the study in this way, we have examined a very specific causal direction. If we wanted to examine the extent to which the reverse causal direction applies (i.e. how likely it is that people who will eventually get cancer are to take up smoking as time goes by), then we would necessarily have to design a cohort study split into "future cancer vs no-future cancer" and measure the uptake of smoking over time.
Update responding to comments:
Note that this is a discussion over a causal direction rather than one of finding a direct causal link. The question of confounding is a separate one. (i.e. there is nothing to suggest that there isn't an independent third variable that both makes you more likely to be a smoker and increases your chances of cancer with time). I.e., in terms of counterfactual causality, we have not definitively shown that "had it not been for smoking these people would not have gotten cancer". But we have shown that "the association between smoking group and cancer would not have increased had time not passed". (i.e. the association is not down to a snapshot of cancer sufferers mere preference for being in the smoking group or not, but is stengthened over time).
A: This is really a question of how to establish causality, because events which are related but not causative WILL likely be correlated in time or space. So looking at some correlated data, how can we determine if the relationship is dependent? A wise research adviser once told me, "correlation does not imply causation, it just tells you where to look".
Let's consider the situation where events A and B are found to be temporally or spatially correlated. If we would like to investigate the preposition that A causes B, the traditional line of thinking is to introduce tests of necessity and sufficiency -- which is what causality really means. 


*

*If the absence of event A leads to the absence of event B, it can be called necessary.

*If only event A alone leads to event B, it can be called sufficient. 


If not having milk causes me to go to the store, what we are saying is not that I get into my empty milk and drive. Absolute causality would mean that whenever I still have milk, I can't be bothered to go to the store; and conversely whenever I'm at the store, it's because I don't have any milk. Now it's easy to see the problem with positively establishing causality in the rigorous sense: most things are not absolutely causal. There's lots of other reasons I might go to the store which aren't related to the milk-state.
This is an easy way to tell a great paper from an alright paper. In careful research, you will see sufficiency and necessity tests everywhere. Making the claim that small-molecule drug A may lead to disassembly of protein complex B? You will immediately see the tests:
necessity

----test----                                ----result----
everything but B                 -->     [nothing] (check for false positive)
everything but A                 -->     assembled
everything with A-like compound  -->     assembled (control group)

sufficiency

A + B alone (in vitro)           -->     disassembled (check for false negative)
A + B + everything               -->     disassembled (trial group)

This is the traditional way you would build an inductive argument for causality experimentally USING correlation, which is what I am confident your lecturer was eluding to!
A: We don't know what the physicist meant.  Two different interpretations follow.

The claim that $X$ preceding $Y$ and being correlated with $Y$ implies that $X$ causes $Y$ is wrong.  It's not enough for $X$ and $Y$ to be dependent even if $X$ precedes $Y$.  For example, $X$ and $Y$ can both be caused by some other variable $W$: $X \leftarrow W \rightarrow Y$.  Or, an even more complicated pattern could arise: $X \leftarrow V \rightarrow Z \leftarrow W \rightarrow Y$ where $Z$ is observed.  Now $X$ and $Y$ are dependent and have no common cause, but neither causes the other.
However, temporal precedence greatly simplifies the conditions for asserting a causal relationship, which you can find in Pearl's Causality book Chapter 2.7 "Local criteria for causal relations".

A variable $X$ has a causal influence on $Y$ if there is a third
  variable $Z$ and a context $S$, both occurring before $X$, such that:
  
  
*
  
*$(Z\; \not\perp\!\!\!\!\perp Y \mid S)$;
  
*$(Z \perp\!\!\perp Y \mid S \cup X)$
  

Essentially, (1) implies that $Z$ is a potential cause of $Y$ given the temporal precedence, and (2) implies that $X$ is able to break that relationship, which can only happen if $X$ causes $Y$.
This condition is much simpler than Pearl's definition for a genuine cause without temporal information.

Another possibility outlined in some of the other answers is that the physicist meant that if $X$ is the passage of time and it's correlated with $Y$, then $X$ causes $Y$.  This statement is correct, but vacuous since the passage of time is the cause of all other variables, by which I mean that the causal graphical structure is this way.  A causal graphical structure is a set of claims about independence relationships given observations and interventions.
A: I speculate that your guest lecturer meant that in physics the only correlations that survive replication are the ones where there is an underlying causal relationship. Time variable is an exception because it is the only variable that is not controlled by the physicist. Here's why.
In physics we usually deal with repeatable phenomena and experiments. As a matter of fact it's almost a given that any experiment is repeatable, and can be replicated by you at later time or by other researchers. So, let's say you observe a sample where $y_i,x_{ki}$ are observations of the variable of interest and independent variables $x_k$. As I mentioned above we fully control the variables $x_{k}$, and can set them to any value we wish.
Your physicist guy is saying that in this setup you will not see any correlation $Corr[y,x_{k}]$ unless there's a causal link. Why? Because someone else or even you yourself will repeat the experiment with any combination and sequence of $x_{kj}$, and only the correlations with causal relationships will survive the replications of an experiment. All other (spurious) correlations will disappear once you collect enough data in all possible combinations of an experiment.
This situation is in stark contrast with social sciences and some business application where you can't do experiments. You observe only one sequence of GDP of a country, and can't change the unemployment holding all else equal and observe the correlations.
Now, time is the only variable that a physicist cannot control. There's only one Jan 1 2017. He can't repeat this day. He can repeat any other variable, but not time. That's why when it comes to time (not lapsed time or age), a physicist is in the same boat as everyone else: correlation does not imply causation for him.
