When spacetime stretches, does both space and time stretch proportionally. That is, if you stretch Planck-length doesn't Planck-time also stretch proportionally? I find this to be a conundrum.
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When space-time stretches, does both space and time stretch proportionally.
No. For example, in the usual form of the Friedman metric, space stretches and time doesn’t. But the question is basically meaningless because in another coordinate system you could have both stretching.
If you stretch Planck-length doesn't Planck-time also stretch proportionally?
This is also a meaningless question. The Planck length and the Planck time are constants. As far as we know, they don’t change, even in an expanding universe.
Added in response to comments:
The Planck length is a unit of length like the meter. The Planck time is a unit of time like the second.
When space expands, as we believe it is doing on a cosmological scale, neither the meter nor the Planck length stretches. Instead, more meters or Planck lengths fit between, say, two galaxies than they did earlier. There is more space, measured in fixed units of length. Similarly, neither the second nor the Planck time stretches. It simply takes more seconds or Planck times for light to travel between those galaxies, measured in fixed units of time.
G. Smith
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