How much energy to mine an ounce of gold?
From the US Mint's financial statement [1]:
2017 cost $13.5 million dollars in "Communications, utilities, and misc charges", which presumably is the bucket that electricity bills falls under. There's no breakdown of overhead between currency coinage production and bullion/numismatic production, but the manufacturing obligation suggests that currency is roughly ⅕ of their total costs, which suggests that we might ascribe about $3 million in utility cost to produce coinage. The coins themselves were worth about $870 million, which suggests about 250 kJ (or 0.07 kWh) per $1 of coinage if we assume the Mint pays $0.05/kWh. I'm not including the energy cost of metal production itself, because I don't want to spend the time to track down that information.
[1] https://www.treasury.gov/about/budget-performance/CJ19/23.%2...
Neither is Bitcoin, so it should be a perfectly valid comparison?
As such, the question is interesting: how much energy required to produce one new ounce of gold? vs one bitcoin?
So I don't think "mining gold" is an appropriate analogy.
EDIT: Case in point: how much did it cost for Visa + Bank to increase a credit-card from $500 credit limit to $2000 credit limit? $1500 of credit appeared out of thin air.
The first credit limit increase "from scratch" costs hundreds of billions of dollars in creating a system of legal compliance, accounting, communication, internal and external risk management, executive compensation, regulatory oversight, credit ratings, cybersecurity, salaries of millions of workers, building public trust in the credit system, writing and passing legislation, even Fed responsibility of building and maintaining a fiat currency.
The second credit limit increase once this system exists is essentially free.
To what extent do these costs need to be accounted for in creating an accurately comparable 'per transaction' metric? By replacing much of the logic with 'just math', Bitcoin has the potential for a much higher theoretical efficiency than the current banking system. Whether market penetration will ever reach the current banking systems levels - and function without replicating these same structures - is certainly highly debatable and for many of the above items implausible.
Bitcoin transactions in isolation are also essentially free - it's the maintenance of the system which keeps it secure and trustworthy that is expensive.