Questions and answers about neo-Newtonian spacetime.

Hi all,

Emma emailed me some questions about neo-Newtonian spacetime, and I thought it would be helpful to post my answers here.   Please feel free to add remarks or follow-up questions in the comments.

--MWP

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1. When you were going through the list of things that might be permitted under neo-Newtonian spacetime, I understood all of the answers except for the last one - if your spaceship is moving 100m/s relative to my spaceship why is that permitted if objects do not have spatial relations through time? Is it because we can manipulate the hyperplanes so that my spaceship could also be moving 100m/s relative to your spaceship? If so, I'm still not sure how 'm/s' is a permissible unit of measurement? To me, in neo-Newtonian spacetime 'metres-per-second' are forbidden or nonsensical.

Answer:  It's not totally obvious, but we can reconstruct relative speed in neo-Newtonian spacetime.  We have simultanaeity, distance at a time, and elapsed time.  So two bodies (spaceships, say) have a distance at one time and another distance at another time.  Lets say they have distance d at time t and distance d' at time t'.  Then over the time interval from t to t' they have an average speed relative to each other, which is just the difference in distance divided by the difference in time:  (d' - d) / (t' - t).  So in neo-Newtonian spacetime we can reconstruct average relative speeds between bodies.
     Notice that we get this "average" without adding up a bunch of values and then dividing by the number of values.  We just take the ratio of the final differences.  In fact, instantaneous speeds are derived from the average speeds by looking at averages over tiny time intervals.  So we can also reconstruct instantaneous relative speeds in neo-Newtonian spacetime.  We just don't have absolute speeds.  Measuring the distance between the spaceships at different times won't tell us which one is moving or which one is moving faster.

2. Is space still absolute in neo-newtonian spacetime? If so, why can't objects move absolutely? I'm just picking up on your line: "Neo-Newtonian space time does not distinguish between absolute velocities." I think I have misunderstood the sentence. Does it mean that an object can move absolutely, but since we have no unit of measurement for velocity in the neonewtonian spacetime, there can be no such thing as a 'difference' in velocity between one object's worldline and another's'?

Answer:  In neo-Newtonian spacetime, space is still absolute in the sense that, within each instant (each spatial hyperplane) there are different places.  There is still an intrinsic difference between being here and being there.  But objects cannot move absolutely because we have here no concept of being in the same place over time.  Motion is change of place over time, and that makes no sense here.  There is no rule (or even an unruly map) to identify a point in space now with a point in space at another time.  So the statement "Neo-Newtonian spacetime does not distinguish between absolute velocities" means that, in neo-Newtonian spacetime, bodies don't even have absolute velocities, and there is not even a distinction between rest and uniform motion.  Quite aside from the matter of stipulating a unit of measure for velocity, we can't even distinguish between some velocity and none at all.  But, as explained above, there is such a thing as a difference in velocity, or a relative velocity.  A body A has a definite relative velocity to any body B, but neither A nor B has a definite velocity on its own.

 3. (continuation from 2) I think my problem is just trying to conceptualise the scenario: if absolute acceleration is permitted, but absolute velocity is not, what is the line moving through as it goes between hyperplanes, if not spatial distance? If I walk from here to the door in a neo-Newtonian spacetime, and am accelerating, my world line will curve upwards through various hyperplanes, yet at the same time, my worldline has not travelled any 'distance'. So here's my attempt to understand it: my worldline is moving through the hyperplanes as if through a series of pictures - so it's more like a dot-to-dot through fixed moments, than a representation of any smooth transition through time resulting in a spatial relation. Is that it?

Answer:  No, a smooth transition is the right picture.  Between any two hyperplanes there are (in the neo-Newtonian model) infinitely many more hyperplanes, forming a smooth continuum.  As we pass from one hyperplane to another, we are always in some hyperplane.  And I think you should think of a curved worldline as representing motion through spatial distance.  It's just that you can't pick out any particular part of the curve as being nearly at rest or as being in very fast motion.  There is no specific distance between temporally separate events, but there are relations between temporally separate events.  The permissible alignments of spatial hyperplanes is constrained by the fact that all inertial worldlines must remain straight.  So you might say there is sort of a loose relation of distance over time, even though it is not fully pinned down.  In that sense, bodies do move through space, just not at any particular speed.
     On the other hand, it might be better to think of bodies as moving through spacetime rather than through space.  The curve traced by a body moves from sheet to sheet in a particular way, even though the distances between points on different sheets are undefined.  It is not moving left-to-right or east-to-west, but forward in spacetime, in a manner that traces either a straight or a curved path.
     I hope that helps, but I think that, with this question, we are dealing with cognitive discomfort more than questions of fact.  You may have to just relinquish certain habits of thought.