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In "An Introduction to Modern Astrophysics" Carroll and Ostlie describe the curvature of space by mass as:

curving in a fourth spatial dimension perpendicular to the usual three of "flat space."

They then add in a footnote:

that this fourth spatial dimension has nothing at all to do with the role played by time as a fourth nonspatial coordinate [their emphasis, but it still doesn't clarify things for me] in the theory of relativity.

However in the spacetime wikipedia it says:

Hermann Minkowski presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. ...Minkowski spacetime is flat, takes no account of gravity... The presence of gravity greatly complicates the description of spacetime. In general relativity... spacetime curves in the presence of matter.

And it also mentions the curvature of time and not space:

Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime... [and] says nothing about curvature of the space component of spacetime.

My confusion seems to be about:

  • Popular explanations separate the four dimensions of spacetime into 3 space and 1 time, so how does curvature in a fourth spatial dimension not lead to understanding spacetime as being 4 spatial dimensions and 1 time dimension?
  • How is a curvature of 3D space interpreted as curving in a fourth spatial dimension while having "nothing at all to do with the role played by time as a fourth nonspatial coordinate"?
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  • $\begingroup$ In (special or general) relativity there is no direct/natural separation of spacetime into space and time. But for any observer there exist an artificial decomposition of spacetime into space and time on a local level in general relativity and on a global level in special relativity. The details can be found in most books on general relativity, for example in the book Semi-Riemannian Geometry by Barett O'Neil (Chapter 6). $\endgroup$
    – jd27
    Commented Jul 7, 2023 at 16:38
  • $\begingroup$ Related: physics.stackexchange.com/a/13839/2451 $\endgroup$
    – Qmechanic ♦
    Commented Jul 7, 2023 at 18:37

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In that paragraph in the book, Carroll & Ostlie use the well known bowling ball in a trampoline model of spacetime. This model has A LOT of problems and is usually only considered only as a way to explain to absolute beginners. Not the least of the problems in their book is that they seem to be saying that the 2D sheet is stretched into 3D, and this implies that normal 3D space is stretched into spatial 4D, and that time is another dimension (perhaps the 5th Dimension Age of Aquarius?)

I find that a much better model is to think that near a planet, space itself is dilated and because of this, time is also dilated. It takes a certain amount of time for light to cross a given amount of space. So if that space is dilated then it takes a longer time to cross it. You have to keep in mind that both measurements are intimately connected. 1 second is the time it takes light to travel 300,000,000 meters. So if space has dilated, then a second must also dilate. Thus we have the warping of spacetime near a planet as used in general relativity.

I prefer to use the term "dilated" rather than "stretched" because it implies stretching in one direction (like the bowling ball example) and in reality space is dilated in all directions and so is time.

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  • $\begingroup$ "space itself is dilated and because of this, time is also dilated" Not to nitpick, but my understanding is that spatial dilation and time dilation can vary independently from one another (unless you add more constraints). In the metric $ds^{2} = -Adt^{2} + Bdx^2 + \cdots$, factors $A$ and $B$ can vary independently. Now in GR, Einstein's equation puts constraints on how $A$ and $B$ can vary, but that's not the same reasoning as what you wrote. Apart from that, everything you wrote seems valid and sound. $\endgroup$
    – MaximusIdeal
    Commented Jul 7, 2023 at 18:28
  • $\begingroup$ @MaximalIdeal the metric is a mathematical model. In the model you can vary the 3 spatial directions in any length or amount that you want, and you can also vary the time dimension in any length or direction that you want. But I don't think that this variability reflects reality. Directions X,Y and Z all grow proportionality to each other, and time also does so. $\endgroup$
    – foolishmuse
    Commented Jul 7, 2023 at 18:43
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    $\begingroup$ Actually, it is mostly that time part is curving, not space. Otherwise, we would have observed a lot of space curvature in the context of astronomy just from Earth's own gravitational pull. Instead, it is a lot of curvature in the time dimension and a little bit in the space parts, where the latter is really just there because you cannot curve time alone, but rather the only spacetime must be curved simultaneously, to satisfy some mathematical requirements. @MaximalIdeal it is thus not independent. $\endgroup$
    – naturallyInconsistent
    Commented Jul 8, 2023 at 4:58
  • $\begingroup$ @naturallyInconsistent Can you lay that out a bit more clearly, mathematically? Something like time dilation = space dilation^2 Keep in mind that it does not take very much time dilation to create gravity. It takes 30,000 years for a one second difference between sea level and the top of Mount Everest. $\endgroup$
    – foolishmuse
    Commented Jul 10, 2023 at 20:47

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