Why you cannot add up speeds

There is no such thing as your speed

Sitting still, you are not moving. You are also travelling at about thirty kilometres a second, and at two hundred and fifty, and at three hundred and seventy. All of those are true at once, and none of them is more true than the others.

This is not a trick or a paradox. Speed is always speed relative to something, and the something has to be named or the number means nothing. Physics has a word for the thing you measure against: a reference frame. Change the frame and the number changes, sometimes by a factor of a thousand. That is not an error bar. It is the answer to a different question.

The first mistake: speeds are arrows, not numbers

Walk down the aisle of an aeroplane at five kilometres an hour while the plane does eight hundred over the ground. How fast are you going? Eight hundred and five, if you walk towards the nose. Seven hundred and ninety-five, if you walk towards the tail. The same two speeds, combined two ways, because direction is part of what a speed is.

Your rotation and the Earth's orbit combine the same way. At local midnight the spin of the planet carries you along its orbital path and the two add. At noon it carries you backwards along it and they partly cancel. Adding the magnitudes — 0.29 plus 29.36 — assumes they always point the same way, which is true twice a day and wrong the rest of the time. The overstatement is only a per cent or two, but it is a per cent or two of something presented as exact.

The second mistake: the galaxy is already in the total

This one is larger. Your motion relative to the cosmic microwave background is your motion relative to the most distant reference the universe offers — and it already contains everything closer in, including the Sun's circuit of the galaxy. They are not two journeys. They are one journey, described from two distances.

Adding them is like taking your speed relative to the aeroplane's cabin, adding your speed relative to the ground, and announcing the total as your real speed. The number that comes out does not correspond to anything. It is not an overestimate of your motion; it is not a measurement of anything at all.

The figure usually quoted for this, six hundred kilometres a second, is not the Sun's either. It is roughly the speed of the Local Group of galaxies against the microwave background. The Sun's own figure is about three hundred and seventy.

So which frame should you use?

Whichever one you can name. That is the whole discipline: state the frame, and the number is meaningful; leave it out, and it is decoration.

The counter on the tool uses the Sun, and says so on the page. Not because it is the largest number — measured against the microwave background you have gone more than twelve times as far — but because it is the widest frame in which every digit comes from an ephemeris rather than a constant known to a few per cent. The galactic figure rests on a speed published anywhere between two hundred and twenty and two hundred and fifty kilometres a second. Running nine ticking decimals off a number that uncertain would be a small lie repeated sixty times a second.

What is actually being measured

Path length, not displacement. You have not ended up thirty-four billion kilometres from where you started — the Earth keeps coming back round, and the whole system moves on. The figure is the distance covered along the way, the reading on an odometer rather than the distance home.

And it is integrated rather than estimated. Earth's orbital speed varies from 29.3 to 30.3 kilometres a second across the year, fastest at the beginning of January when it is nearest the Sun. Multiplying an average by an elapsed time gets the galactic figure wrong by a fraction of a per cent, which sounds like nothing until you notice that a fraction of a per cent of that number is several billion kilometres.