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In some discussions, people seem to imply that spacetime is in an actual ellipse around a massive object so that, for example, the planet orbiting a star is actually traveling in a straight line in spacetime as it orbits the star. But my understanding is more that spacetime just curves slightly around the star and that the planet is simply falling down a slope towards the star as the planet attempts to fly off in a straight line.

It seems to me that if spacetime was in an ellipse then everything, including light would get trapped in the ellipse and never escape, just like a planet. Or perhaps the ellipse concept is correct and the result has more to do with the speed of the orbiting object. This would be the time part of spacetime.

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    $\begingroup$ Your (and my own) physical pictures will eventually let us both down, nobody can visualise 4 dimensions. Only mathematics can give you a correct description. $\endgroup$
    – user198207
    Commented Jul 3, 2018 at 17:47
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    $\begingroup$ Is spacetime in an ellipse... What do you mean? Spacetime is 4D. An ellipse is only 1D. $\endgroup$
    – Qmechanic ♦
    Commented Jul 3, 2018 at 19:24
  • $\begingroup$ @Qmechanic: I think my original question explains what I mean. Is a planet, orbiting around a star, traveling in a straight line in spacetime? A straight line that circles around the star. Or, to the planet, is a straight line still outwards from the star but it keeps falling down the spacetime slope towards the star. $\endgroup$
    – foolishmuse
    Commented Jul 3, 2018 at 20:38
  • $\begingroup$ The closest things to straight lines in a curved space are trajectories of light. They are "still outwards", not elipses around a star. Planets on elliptical orbits do not move along straight lines. There is one exception though. When gravity is very strong, such as around a black hole or large neutron star, light can indeed move on an orbit around the star. The sphere of these orbits is called the photon sphere. If our Sun were squeezed to become a neutron star, the diameter of its photon sphere would be 9 km. So a circle 9 km in diameter around the squeezed Sun would be a straight line. $\endgroup$
    – safesphere
    Commented Jul 3, 2018 at 21:04
  • $\begingroup$ @safesphere: So we can confidently say that light travels in a straight line in spacetime? Whether that be coming from a distant location and curving between galaxies, or even traveling in an unending circle, trapped around a black hole or neutron star? $\endgroup$
    – foolishmuse
    Commented Jul 3, 2018 at 22:39

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When you say that "spacetime is an actual ellipse" around a massive object, what it really means is that the massive object creates curvature in spacetime such that geodesics near the object form the path of an ellipse (an orbit). The actual "shape of spacetime" around the object is decidedly not an ellipse -- it's a four-dimensional shape that can only be described mathematically, not visualized. (The standard visualization is a ball lying on a rubber sheet, but this shouldn't be taken too literally.)

The amount of curvature depends on the density of mass (and/or energy) in a given volume.

You are correct that a star creates a relatively "slight" curvature in spacetime, allowing planets to orbit around it, but allowing light to escape easily. However, if the star were to collapse into a black hole, the curvature would become much greater as you approach it, and there would indeed be places near the black hole where photons can orbit around it (the photon sphere), although these orbits would be very unstable.

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  • $\begingroup$ When you say "if the star were to collapse into a black hole" Do you mean that a black hole, the same mass as the former star, would create greater curvature than the star, even though the mass is the same? Or do you mean that as the star attracts more and more objects, and eventually collapses into a black hole because of the greater mass, then it will cause greater curvature because of the greater mass? In another discussion, I was told that a large gas cloud has the same gravity as a star made up from that cloud. $\endgroup$
    – foolishmuse
    Commented Jul 3, 2018 at 18:37
  • $\begingroup$ @foolishmuse The black hole can be the same mass as the star. The key is that the mass of the black hole is confined to a much smaller volume than the star, and this is what creates greater curvature. $\endgroup$
    – Dmitry Brant
    Commented Jul 3, 2018 at 18:40
  • $\begingroup$ Further research on the density issue led me to this link: quora.com/Is-gravity-a-function-of-density-or-mass $\endgroup$
    – foolishmuse
    Commented Jul 3, 2018 at 20:41

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