Is weak sustainability (Hartwick's rule) possible on a global scale? Hartwick's rule postulates that an economy is sustainable if all rents obtained from the extraction of non-renewable resources are re-invested in produced capital (Source). A typical example would be the Norwegian Sovereign Wealth Fund that invests the resource rents obtained from oil into equity, real estate and fixed income. (Source)
In generic terms the investments that the fund makes will be in capital (e.g. machinery) or land and other rent-generating assets (e.g. land, patents, etc.). However, the produced capital will depreciate over time and the rent-generating assets do not involve the production of a new asset but the purchase of the rent attributable to the rights over an existing asset.
How then would weak sustainability (Hartwick's rule) be possible on a global scale? Presumably this would have to involve investing every cent of resource rent into capital investments with a lifespan that is at least as long as the lifespan of the resource that it substituted?
First of all Hartwick's rule has originally been derived under rather specific assumptions, among others a Cobb-Douglas production function and constant population and technology. Therefore copying it one on one to the real world would require a huge leap of faith.
Although it has later been generalized to broader classes of production functions, people have also pointed out that instead of a prescriptive rule (invest resource rents in man-made capital to realize constant consumption), it is in fact a descriptive result. If we observe constant utility then resource rents must have been reinvested.
Asheim, Buchholz and Withagen (2003) show among others that it is perfectly possible in a world with constant technology and population not to have zero net investment, and still constant utility over time, as well as situations where we follow the Hartwick rule and have non-constant consumption (and hence non constant utility).
Blindly following the Hartwick rule, on a global scale or on a regional scale for that matter would therefore not be a good idea. That notwithstanding several papers have shown strong links between savings in the past allowing for constant or even growing consumption in the (far) future (e.g. here, here and here).
Intuitively it makes sense that saving rents from non-renewable resources to make-up for lost production capacity is a good policy measure. Whether or not it will be enough is a different matter, and very much depends on degree of substitutability and technological progress.