B Question about bendability of objects in 4 dimensions

[PeterDonis]

a permanent slowing of the earth’s spin by more impact than this discrepancy which leaves me to my point, I think our timekeeping is quite incredibly precise for geocentric purposes

Um, you do realize that the reason we know about the change in the rate of the Earth's spin on such short tlime scales is that we have clocks that are much more accurate than the difference, right?

The issue is not the Earth's spin per se. It is the difference in "rate of time flow" of clocks at different places on Earth. We have clocks that can detect that difference even over differences in height as small as about one millimeter.

 

[PeterDonis]

So then when you actually get to 0.8c you stop accelerating at a constant rate you then unbend ?

Your worldline does. In other words, your "path through spacetime" is bent while you are accelerating (i.e., feeling a force) and straight when you are not.

 

[ESponge2000]

You do realize that "sea level" is about 13 miles higher (i.e., further from the center of the Earth) at the Earth's equator than it is at the poles, right? And yet clocks still run at the same rate everywhere at "sea level"?

Well an atomic clock can be set somewhere in London and a formula to advance clocks 0.04 hundredths of a second or some surface distance formula can most entirely do the trick… and if the clock is ported in London then it’s perpetually rescyncronized to correct the departures from London time, which is actually how smartphones keep time I think … when online it resyncs to an atomic control clock and when offline it uses analog methods to keep time, gearing off the last resync. I think laptop clocks work this way as well. This isn’t I think , I’m pretty sure this is the way electronics have been working for a decade now

 

[ESponge2000]

Your worldline does. In other words, your "path through spacetime" is bent while you are accelerating (i.e., feeling a force) and straight when you are not.

Well that is obvious but how does it look during the constant acceleration , x axis slopes upward or is it concave down?

 

[PeterDonis]

a formula to advance clocks 0.04 hundredths of a second or some surface distance formula

I'm not sure what you mean here.

if the clock is ported in London then it’s perpetually rescyncronized to correct the departures from London time, which is actually how smartphones keep time I think ... when online it resyncs to an atomic control clock and when offline it uses analog methods to keep time, gearing off the last resync

Smartphones don't sync directly with atomic clocks, they usually get their time from whichever cell tower they are connected to. Some might also check NTP servers, which is how most laptop and desktop computers get their time.

However, these sync operations have nothing to do with any relativistic effects. Smartphone clocks, and indeed any ordinary clocks, aren't accurate enough to detect those. The sync operations are actually because ordinary clocks, whether they're in smartphones, computers, bedside alarm clocks, kitchen appliances, or whatever, simply don't keep time very well at all. Their "tick rates" are so variable that they have to be reset at periodic intervals.

Actual atomic clocks that are accurate enough to detect relativistic effects are at fixed locations on Earth (unless they're being used for a special experiment like the Hafele-Keating experiment), and keeping them synchronized is just a matter of periodically having them exchange signals of some sort to check their timekeeping. The difference in their "tick rates" due to altitude relative to the geoid is already known to great accuracy and can be adjusted for locally at each clock.

 

[PeterDonis]

how does it look during the constant acceleration , x axis slopes upward or is it concave down?

Your acceleration doesn't change the $x$ axis of your coordinates.

Your worldline is concave in the direction you are accelerating. For example, if we just consider the $t$ and $x$ axes, the worldline of an object with constant proper acceleration in the positive $x$ direction is a hyperbola that is concave to the right (the positive $x$ direction). The asymptotes of the hyperbola are the 45 degree "light cone" lines through the origin.

 

[Dale]

Sort of but the part about constant acceleration preserves the length I don’t completely grasp that

Lengths will be distorted when the internal stresses change. If the acceleration is constant then the internal stresses are also constant.

 

[DrGreg]

Your worldline is concave in the direction you are accelerating. For example, if we just consider the $t$ and $x$ axes, the worldline of an object with constant proper acceleration in the positive $x$ direction is a hyperbola that is concave to the right (the positive $x$ direction). The asymptotes of the hyperbola are the 45 degree "light cone" lines through the origin.

Here is a spacetime diagram of a rod accelerating along its own length.

The green area is the rod. This is drawn under the assumption of constant proper acceleration and a Born rigid rod.

On the diagram below I have superimposed some snapshots of the rod at various times, using the definition of simultaneity of a single inertial frame.

And on the diagram below I have superimposed some snapshots of the rod at various times, using the definition of simultaneity of an accelerating frame in which the rod is at rest.