Yes, it's independent of the absolute level of debt. A 500 trillion dollar debt means the private sector is proportionately awash in 500 trillion of liquid assets. This will be reflected in a proportionately high (dollar-denominated) GDP, tax revenue, and consumer price level (absent any big changes in population or productivity).
Of course, to get from 40 trillion to 500 trillion would mean that prices are basically 12x higher, which would mean a lot of inflation will have happened in the meantime. So it would be very bad if the government debt increased by that much in the timespan of one year, because it would basically mean hyperinflation over that span of time. But if that same growth in the debt happened over 400 years, it would be no big deal.
So the relative rate of growth of the government debt certainly matters, because that influences inflation, which is the thing that actually causes problems. Not the size of the debt itself. That is, if G is outstanding government debt, then the figure that matters for inflation is approximately (1/G) dG/dt. But not the absolute level of the government debt G itself.
This also means that compounding interest doesn't really affect the calculation. As the debt grows larger, and the interest payments grow larger, directly in proportion to the size of the debt, and therefore the economy as a whole -- they don't outgrow it. Assuming a steady and reasonable interest rate, at least. If the interest rate were super high or growing without bound, then yes, that would be a problem for the government debt. But that would be a pretty weird thing to happen and wouldn't happen just because the debt figure itself hit some large value, but probably instead because of a currency crisis (eg. the country owes debts to other countries in currencies it does not control).
I'm trying to think of a good way to put this. A person can run out of water and die of thirst, but when you zoom out to bigger and bigger scales, the Earth itself doesn't run out of water; it just goes around in a cycle. Economists have a saying that "one man's expenses are another man's income". For a single household, that doesn't really feel true; the rest of the economy is so big that expenses bascially just disappear from your bank account, and you don't notice any of the money that leaves your pocket when you hire a plumber come back to you, even though some tiny amount actually does when that plumber buys food from your restaurant. So we individuals also can have the experience of debts growing big enough to bankrupt us as the interest payments exceed our income. But governments live on the same scale of economies as a whole, and for them, the recirculation of the money they spend really can't be removed from the analysis.
It comes directly from the math of double entry bookkeeping.
In my analogy, individual people do need to take care to ensure they don't run out of water. But you don't need to worry that if it rains a lot this year, there won't be any rain left for your children.
Right. It's not existential, sure. It helps that all our debt is issued in our currency. But the fact is that ~30% of our national debt is held by foreign entities, and at the very least interest payments are an outflow of wealth from our country. This is not a healthy position if we value freedom of action. It WILL eventually constrain our country in meaningful ways. At the very least the mechanisms to manage this debt will weaken our currency, leaving everyone that doesn't invest in the market behind (which is a sizeable portion of our citizenry).
Pointing out that earth is a closed system so it's all good doesn't address these very real concerns about our unchecked national debt.
Of course, to get from 40 trillion to 500 trillion would mean that prices are basically 12x higher, which would mean a lot of inflation will have happened in the meantime. So it would be very bad if the government debt increased by that much in the timespan of one year, because it would basically mean hyperinflation over that span of time. But if that same growth in the debt happened over 400 years, it would be no big deal.
So the relative rate of growth of the government debt certainly matters, because that influences inflation, which is the thing that actually causes problems. Not the size of the debt itself. That is, if G is outstanding government debt, then the figure that matters for inflation is approximately (1/G) dG/dt. But not the absolute level of the government debt G itself.
This also means that compounding interest doesn't really affect the calculation. As the debt grows larger, and the interest payments grow larger, directly in proportion to the size of the debt, and therefore the economy as a whole -- they don't outgrow it. Assuming a steady and reasonable interest rate, at least. If the interest rate were super high or growing without bound, then yes, that would be a problem for the government debt. But that would be a pretty weird thing to happen and wouldn't happen just because the debt figure itself hit some large value, but probably instead because of a currency crisis (eg. the country owes debts to other countries in currencies it does not control).
I'm trying to think of a good way to put this. A person can run out of water and die of thirst, but when you zoom out to bigger and bigger scales, the Earth itself doesn't run out of water; it just goes around in a cycle. Economists have a saying that "one man's expenses are another man's income". For a single household, that doesn't really feel true; the rest of the economy is so big that expenses bascially just disappear from your bank account, and you don't notice any of the money that leaves your pocket when you hire a plumber come back to you, even though some tiny amount actually does when that plumber buys food from your restaurant. So we individuals also can have the experience of debts growing big enough to bankrupt us as the interest payments exceed our income. But governments live on the same scale of economies as a whole, and for them, the recirculation of the money they spend really can't be removed from the analysis.