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If I'm standing on a weighing scale on the surface of the Earth, both I and the scale have the same proper acceleration of $g$ upwards, yet the scale reads non-zero.

But if both I and the scale are undergoing the same proper acceleration (say $g$) in empty space, then the scale will read $0$ (rather I won't be able to stand on the scale in any meaningful sense).

What's the difference? In both cases our proper accelerations are the same, yet there's zero reading in one and non-zero reading in the other scenario.

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The reading of the scale depends on the normal reaction force between the scale and your body .

In the first case , the earth exerts a downward force on you . Hence , to accelerate upwards there needs to be a force acting in the upward direction to counter gravity. This is the normal reaction force exerted by the scale on you ( and by newton's third law , you exert the same amount of force on the scale). Hence , we get a non zero reading on the scale ( as reading depends on the normal reaction force)

In the second case , as you are in empty space there's no gravity. But since you are accelerating , there needs to be a force acting on you . The only force on you is the normal reaction in this case . So I believe that you would still get a non zero reading on the scale ( as the normal reaction is not zero ).
But the reading of the scale in case 2 is gonna be lesser than case 1 ( infact its half of the reading in case 1) as we have no gravity.

Also , I think we would have no problems standing on the scale . The situation is gonna be similar to standing on the ground and weighing yourself on earth.

Infact , this is the key idea behind einstein's theory of general relativity. Whether we are on earth or on a body accelerating in empty space , the observations are gonna be the same ! .

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