- #26

- 3,561

- 1,342

The big problem is where you state that you will ignore the front and rear potential time difference in the spaceship. You can't make a comparison to gravitational time dilation at the Earth's surface without taking into a account this potential time difference, Because gravitational time dilation is a measurement of clocks at different potentials. To make any comparison, you would have to do the following type of experiment:Some time ago there was a similar thread

https://www.physicsforums.com/threa...me-dilation-and-equivalence-principle.929838/

but what I want to discuss is similar but not the same and I would like to specify my question in such way, that it hopefully wont go sideways as in cited previous thread.

So we have two labs.

First lab is on surface of Earth, time dilation is equivalent to gravitational potential of surface of Earth, gravity is 1 g.

Second lab is on spaceship which is lets say 1 light year away and there is no star or planet around. The starship accelerates with proper acceleration 1 g and its speed relative to Earth is quite low, lets say 0,00001 % of speed of light. So time dilation relative to Earth caused by speed is close to zero. Due to relativity theory, acceleration causes no time dilation, so there all together there is no time dilation on the spaceship.

Now both labs have equipment (accelerator?) which can produce muons. Muons are created in both labs and they decay in time t1 in first lab (on Earth) and time t2 in second lab (in spaceship). Based on my understanding of relativity, gravitational time dilation on Earth is very slightly slowing down the decay of muons on Earth and therefore time t1 is bigger than time t2 on spaceship, where is no time dilation and muons decay faster than on Earth.

Just to make the example precise, muons dont jump up and down in the ship, so I will just ignore front and rear potential time difference in the spaceship.

So finally, based on Equivalence principle there should be no measurable differences between proper acceleration and gravity, but in the described example the scientist which knows how much time it takes for muon to decay on Earth can distinguish if the lab is on Earth or inside a spaceship.

What is wrong in my statement that you can use muons to measure difference between acceleration and gravity?

At the Earth, you have two identical clocks. One sitting on the surface and one 10 meters above it. On the ship, you have two identical clocks, one at the rear and one 10 meters closer to the nose. You then compare how much the clocks on Earth differ after the lower clock has ticked off some interval to how much the clocks on the ship differ after the rear clock ticks off the same interval.

This difference will not be the same, it will turn out that the clocks on the ship will differ from each other just a tad more than those on the Earth. This is because the potential between the clocks is the equivalent of that for a gravity field that maintains a constant 1g over the distance separating the clocks, while on the Earth, gravity falls off slightly between the two heights of the clocks. Thus you get a larger potential difference in the ship than you do for the Earth.

This does not violate the Equivalence principle however. For example, what if I have another two clock, again 10 meters apart, but this time the lower clock is twice the radius of the Earth away from a planet with 4 times the Earth's mass. That lower clock would be at 1g of gravity just like the Earth surface clock is. However, since 10 meters is a smaller fraction of 2 Earth radii than 1 Earth radius, gravity will fall off less between these two clocks, and the potential between them will be larger than that for the Earth clocks, and thus the difference in tick rate between this pair will be larger than for the Earth pair. In both cases the differential tick rate is caused by gravitational time dilation, and even though the lower clock of each pair is experiencing the same force of gravity, the differential between of the clocks of each pair is different.

You will also notice that our new pair of clocks more closely matches the clocks in the spaceship than the Earth clocks do. If we increase their distance from the planet while simultaneously increasing the mass of the planet in order to maintain 1g at the lower clock, the behavior of these clocks become closer and closer to that of those in the ship. The behavior of these two set of clocks begin to converge. At some point they will become all but indistinguishable. This is the main point of the equivalence principle: that

*over a small enough region*gravity and acceleration are equivalent.