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So, let's say that we have an incredibly long, virtually indestructible ruler. We have advanced enough to move it wherever we want. Let's also say that we have another, identical ruler of the same length and other properties. If we put one of these rulers close to a black hole (remember, it's a really tough ruler) and another one parallel to it far away from the black hole (in empty space for our purposes), what would happen to the rulers with respect to one another? Let's say that the ruler next to the black hole is also suspended by a spaceship and is therefore not being sucked into the black hole. Would the endpoints of one ruler be closer together than the other? Would each 'tick' in the ruler in the gravitational field be closer to one another compared to the other ruler? I thought of this after reading about gravitational time dilation and I was curious about a length contraction (spacelike) analog. I am also just curious about just what happens to space in a gravitational field (using a ruler as a stand-in for space for the sake of intuition). Finally, to clarify, this is a non-charged, non-rotating black hole. Feel free to explain what would happen in these scenarios, but just a static, non-charged black hole is fine for a basic explanation.

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  • $\begingroup$ What about the fact the black hole curves space time? And how are you measuring it? Placing another ruler beside it? Aiming lasers from outside the black hole at each end and triangulating? $\endgroup$
    – DKNguyen
    Apr 8, 2021 at 3:06
  • $\begingroup$ Vacuum solutions are Ricci flat meaning the volume of small spacetime balls is unchanged. So when time is dilated, length is radially contracted in the exact same proportion, as observed remotely. Your question is unclear, because "closer together" depends on who is measuring, where, and how. $\endgroup$
    – safesphere
    Apr 8, 2021 at 4:53

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There's no length contraction in this situation. Length contraction involves setting up a coordinate system and comparing the proper length of an object to a coordinate length, but there's nothing to measure coordinate length here. The rulers are far apart and each one measures only its proper length.

A ruler suspended from above in a gravitational field will stretch, and in a stronger field it will stretch more, but as far as general relativity is concerned (ignoring everything we know about realistic properties of solids), the amount of stretching can be made arbitrarily small by increasing the Young's modulus of the material. So an idealized ruler can just measure metric distance (plus epsilon), even close to a black hole. You have to give physical meaning to the notion of metric distance somehow, and this is one way to do it.

Gravitational time dilation can be observed by having a clock lower in the field emit a light signal at its natural oscillation frequency, which is compared to the frequency of an identical clock higher up when it arrives there. You could, analogously, use the ruler close to the black hole as an antenna to emit a light signal with a wavelength proportional to its length, and measure the wavelength far from the hole. It would be redshifted and therefore larger than the proper length of the other ruler. But I don't think it's reasonable to conclude from that that the ruler near the black hole is longer. It is, in any reasonable sense, the same length as the ruler far from the hole (up to epsilon), and the redshift is due to gravitational time dilation.

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Assume you measure the length of each ruler by how long it takes a photon to travel from one end to the other. Due to time dilation near the black hole, the photon (as measured far away from the black hole), would take longer to traverse the ruler. This can be interpreted as the ruler near the black hole being longer (space expanding) or equivalently as time slowing down.

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