# How does one measure the world?

Suppose one took our universe and divided everything in half; mass, space, time and so on - would we notice? And if we don't doesn't this mean that there is no intrinsic notion of length - despite appearances?

Or should I say absolute?

• Every measure is relative to a reference we only ever measure relations. The choice of units is somewhat arbitrary. – Quentin Ruyant Apr 5 '15 at 8:57

I don't think you can choose a way to do this that would allow the speed of light 'c', the Planck constant 'h-bar' and the gravitational constant 'G' to all remain the same.

It has always seemed to me that the balance of these three is what sets the scale of the universe. The ratio of the first two vaguely governs how mass and speed are related in one way, and the other determines how mass and speed are related in another way. The odds of these balancing out at multiple points seems vanishingly small.

So let's try:

If we divide both length and time in half we have maintained c.

You have divided times in half, so h-bar, which limits the product of energy and time, would go down to half of its original value unless you double energies, which requires you double masses (given E=mc^2).

Then what happens to the gravitational constant? Everything weighs twice as much and is half as far apart, so the force between two things will be (M + m)G/r^2, which has increased eightfold if you double M and m, and halve r (which doubles the value, twice). So G would have to change significantly.

If the arithmetic is correct, I think this implies that however you pulled this off, we would notice -- one of the three values would change.

• Ok; good answer. I suppose the only possibility then is to try non-linear changes. – Mozibur Ullah Apr 5 '15 at 9:18
• Wouldn't any non-linear adaptation upset the various inverse-square laws pretty fast? – user9166 Apr 5 '15 at 13:18
• I think you have to accept that physics as we know it still has an intrinsic sense of scale. This is one of the big gaps between Relativity and Quantum Dynamics. Relativity messes with the sense of scale that quantum numbers demand in highly unrealistic environments. Since we can more directly measure the quantum effects, we need a small modification of Relativity that we have not yet found. – user9166 Apr 5 '15 at 13:20
• Well there is such a thing as conformal field theories; which are QFTs; they have scale-invariance; one use is in string world-sheets; but the argument above, really is one way of thinking about general covariance in relativity - so only not the world in the small. – Mozibur Ullah Apr 5 '15 at 15:22
• From what I recall now of physics, the fine-structure constant is the only pure constant ie without units; so I suppose that sets a scale. – Mozibur Ullah Apr 5 '15 at 15:26