No, it's just round in the sense of a beachball, say. Gravity can curve light, so making things seem different in shape - but any curvature due to gravity is very subtle for the Earth. (For a very massive object like a black hole you have some curvature you'll see due to gravitational lensing, but for the Earth, the gravity is so small, light follows an almost perfect straight line).
Instead, the curvature of space-time is noticeable when you move things. When you bring in time, it's not subtle at all, it is an easily noticeable curvature.
If you throw a ball, the reason it follows an arc, not a straight line, is because of curved space time. It's not space itself though that's curved, it's space and time together in general relativity.
The ball is actually following the longest path it can find in space time - not the shortest path as some mistakenly say. It spends as much time as it possibly can to get to its destination. It follows the path of "greatest proper time" - the idea is that a clock that moves along that path measures the greatest possible time interval out of all the paths joining those two points. Or, to put it another way, if your clock, say, ticks every millisecond, then it follows the path which gives it the maximum possible number of ticks. A more informal way of describing this is the "principle of maximum aging" - objects in free fall, according to general relativity - follow paths that let them age as much as possible.
So, for instance, going all the way to the Moon and back would mean it gets there more quickly in its own subjective time because of relativistic time dilation. Moving in a straight horizontal line also takes a bit longer because of the gravitational time dilation, that clocks tick more slowly in a gravitational field. You want as many ticks as possible for the maximum proper time. So if you throw a ball, in such a way as to hit a point at the same height above the ground some distance in front of you, it needs to follow a curved path where it rises high enough for its clock to speed up a bit. But rise too high, and its clock slows down instead of speeding up because it is moving so quickly.
So the curved flight of a thrown ball is a compromise between these two effects.
Described better here: The Principle Of Maximal Aging
I tried a search for a good description. Many were simply incorrect. This is a widely misunderstood topic. In most areas of maths a "geodesic" is a curve of shortest length according to some metric. But in general relativity, it is a path of longest length according to the proper time metric.
Others were far too technical. The articles in wikipedia, though accurate in as far as I understood them (whuch is not much - I'm not at all specialist in this area), were unreadable by anyone not specialist in this topic area. That one though has the balance right, reasonably well explained without any maths and also accurate.
So - the curvature involves geodesics of maximum proper time, or maximal aging. Space then isn't curved, it's rather, space and time together.
I've never studied general relativity as a mathematician, and have only a lay person's understanding of it, am only answering this because so many answers on the web, and also some answers to other related questions here on quora get it all back to front. That article though is good and should help you get it straight.
This goes into it in more detail: Page on astronomycorner.net
This is all according to General Relativity. It's important I think to realize though that GR, though the best we have in some ways, is also known to be incomplete and incorrect, strictly speaking. It can't be made compatible with Quantum mechanics, and though it does a wonderfully good job of description of motion of masses, it has no explanation at all of what matter is, why it has mass, and why it curves space-time. Joining those two together, the matter, and space time curvature, is a task for the future, with many ideas and theories but nothing proved. And, though most would think I think that the result would still use space time curvature, who knows, maybe some other notion that has similar predictions would take its place. GR provides extremely accurate predictions so any successor theory would need to at least match those, to many decimal places in some cases, to improve on it.
Instead, the curvature of space-time is noticeable when you move things. When you bring in time, it's not subtle at all, it is an easily noticeable curvature.
If you throw a ball, the reason it follows an arc, not a straight line, is because of curved space time. It's not space itself though that's curved, it's space and time together in general relativity.
The ball is actually following the longest path it can find in space time - not the shortest path as some mistakenly say. It spends as much time as it possibly can to get to its destination. It follows the path of "greatest proper time" - the idea is that a clock that moves along that path measures the greatest possible time interval out of all the paths joining those two points. Or, to put it another way, if your clock, say, ticks every millisecond, then it follows the path which gives it the maximum possible number of ticks. A more informal way of describing this is the "principle of maximum aging" - objects in free fall, according to general relativity - follow paths that let them age as much as possible.
So, for instance, going all the way to the Moon and back would mean it gets there more quickly in its own subjective time because of relativistic time dilation. Moving in a straight horizontal line also takes a bit longer because of the gravitational time dilation, that clocks tick more slowly in a gravitational field. You want as many ticks as possible for the maximum proper time. So if you throw a ball, in such a way as to hit a point at the same height above the ground some distance in front of you, it needs to follow a curved path where it rises high enough for its clock to speed up a bit. But rise too high, and its clock slows down instead of speeding up because it is moving so quickly.
So the curved flight of a thrown ball is a compromise between these two effects.
Described better here: The Principle Of Maximal Aging
I tried a search for a good description. Many were simply incorrect. This is a widely misunderstood topic. In most areas of maths a "geodesic" is a curve of shortest length according to some metric. But in general relativity, it is a path of longest length according to the proper time metric.
Others were far too technical. The articles in wikipedia, though accurate in as far as I understood them (whuch is not much - I'm not at all specialist in this area), were unreadable by anyone not specialist in this topic area. That one though has the balance right, reasonably well explained without any maths and also accurate.
So - the curvature involves geodesics of maximum proper time, or maximal aging. Space then isn't curved, it's rather, space and time together.
I've never studied general relativity as a mathematician, and have only a lay person's understanding of it, am only answering this because so many answers on the web, and also some answers to other related questions here on quora get it all back to front. That article though is good and should help you get it straight.
This goes into it in more detail: Page on astronomycorner.net
This is all according to General Relativity. It's important I think to realize though that GR, though the best we have in some ways, is also known to be incomplete and incorrect, strictly speaking. It can't be made compatible with Quantum mechanics, and though it does a wonderfully good job of description of motion of masses, it has no explanation at all of what matter is, why it has mass, and why it curves space-time. Joining those two together, the matter, and space time curvature, is a task for the future, with many ideas and theories but nothing proved. And, though most would think I think that the result would still use space time curvature, who knows, maybe some other notion that has similar predictions would take its place. GR provides extremely accurate predictions so any successor theory would need to at least match those, to many decimal places in some cases, to improve on it.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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