While this might seem like a trippy question, the consequences are very real.
The notion derives from a couple of ideas - primarily that it's possible to disconnect economic growth from environmental impacts.
GDP can climb infinitely inside an economy that uses no net resources and doesn't pollute. Unfortunately there is no evidence to support this and, if it's possible, it hasn't happened yet.
The second idea is so-called "cloud computing" which has the superficial appearance of enabling a weightless economy.
The concept is very simple: rather than running software on your computer at home or in your office, you use a service provided over the internet. If you use Hotmail, Gmail or Dropbox, that's cloud computing.
It's "cloud" because it's happening "out there" - it could be any location on the planet with internet access.
Because running the systems is somebody else's problem, it feels invisible and therefore "weightless".
Ultimately, however, physical boxes and wires are involved, and nothing comes for free. Online data centres require continual upgrades and replacement, which naturally consume resources and contribute to growing mountains of e-waste.
They consume prodigious amounts of energy, with predictions that by 2025 the technology sector could consume 20 per cent of the world's total electricity.
Currently it accounts for 6-10 per cent of consumption, which equates to 4 per cent of global greenhouse gas emissions.
The largest data centres require nearly 100MW, roughly a tenth of the output of a thermal power station. A single web search is responsible for about 0.8g CO2 equivalent.
A major part of their energy consumption is their need for cooling, which is why some are now located in Iceland where they can be powered by renewable sources. Companies such as Google and Amazon are actively reducing their emissions by using solar and wind energy.
The problem is not cloud computing per se. In fact, there are good reasons why cloud computing is preferable.
Economies of scale and related technologies allow more flexible use of resources. Computing power can be rapidly shuffled between systems to balance the load, making it more flexible and efficient.
Large-scale data centres can recover and reuse heat. A Google funded research paper suggests that cloud computing might reduce energy consumption by 87 per cent.
Even in the cloud, however, demand continues to grow.
The cryto-currency bitcoin, for example, needs huge amounts of electricity.
In July 2021, a single transaction used more than 1700 kilowatt hours of electricity, making it equivalent to about 59 days' worth of power consumed by an average US household.
Ultimately, the issue can be attributed to the rapid rate of growth as more people use more computing.
As long as this continues there will be no weightless economy.
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