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I am new to physics and I have learned a little bit about gravity from Einstein's perspective. The gist is that heavy objects create curvature of spacetime, and free-falling objects move on the straight lines in the curvature.

But I am failing to understand how this applies to objects on Earth. For example, why don't elephants make spacetime curvature and cause dust to go around them? Or simply, how does spacetime curvature work inside a planet?

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    $\begingroup$ You could ask the same question about gravity from Newton's perspective. Do you understand the answer in that case? $\endgroup$ Apr 12 '20 at 20:10
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    $\begingroup$ In Newton's case matter attract each other. and which explains from his equation and the heavy objects (which is earth) attract the other objects and compare to the earth the attraction of other objects is insignificant. $\endgroup$ Apr 12 '20 at 20:12
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    $\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$
    – David Z
    Apr 13 '20 at 5:12
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    $\begingroup$ As is often the case with questions of this sort, you've made it harder to ask the question by phrasing it as a question about something counter to fact. Relativity does seem to work on the surface of the Earth for the simple reason that it works everywhere, so asking why it does not is asking a question for an explanation of a falsehood. In the future, try asking questions more like "how could we observe the effects of general relativity on ordinary objects the size of an elephant or a dust mote?" or similar question that does not presuppose a falsehood. $\endgroup$ Apr 15 '20 at 3:07
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    $\begingroup$ @EricLippert You analyzed and worded very nicely how such a question should be understood and answered by a benevolent person ;-). $\endgroup$ Apr 15 '20 at 18:06
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The short answer: general relativistic effects are mostly not noticeable on such a small scale (except in a few cases). For example, one common way to characterize the strength of a gravitational field is through the dimensionless number: $$A=\frac{GM}{Rc^2}$$ where $G = 6.67 \times 10^{-11} \text{ }\mathrm{m^3 kg^{-1} s^{-2}}$ is the gravitational constant , $M$ is the mass, $R$ is the distance to the object, and $c=2.99\times 10^8\text{ m/s}$ is the speed of light. For being directly next to an African Bush Elephant: $$A \approx 10^{-24}$$ For the Earth: $A \approx 10^{-9}$ and even for the Sun, $A \approx 10^{-6}$. Generally, when $A \ll 1$ the effects are practically negligible. In fact, most of the physics that describes planetary motion in our solar system can be accurately described with standard Newtonian physics. However, there are some extremely sensitive cases (e.g. orbit of Mercury / satellite communication) where we need to take into account general relativity.

Just for comparison, a black hole (using the Schwarzschild radius) gives a value of $A=0.5$, which is much stronger than our Sun's meaning we definitely will need to take into account general relativity.

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    $\begingroup$ For that matter, even most Newtonian gravitational effects aren't easily noticeable on the Earth's surface. Mice and small rocks don't orbit elephants. For pretty much all everyday purposes, plain old Galilean gravity ("all massive objects fall downwards with the same acceleration") is more than sufficient. $\endgroup$ Apr 13 '20 at 6:51
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    $\begingroup$ While I do not agree with @DoctorNuu statement, he has a point in his answer. So you cannot say "not noticable", but rather "in most cases not noticable" or similar. GPS is in fact a good example for where it is perfectly well valid. $\endgroup$
    – Mayou36
    Apr 13 '20 at 8:44
  • $\begingroup$ I've deleted a number of inappropriate comments and/or responses to them. $\endgroup$
    – David Z
    Apr 13 '20 at 9:10
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    $\begingroup$ @IlmariKaronen Now I want to read a story about a world where mice and small rocks orbit elephants. $\endgroup$
    – user253751
    Apr 13 '20 at 11:28
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    $\begingroup$ @IlmariKaronen : I wouldn't really call it "Galilean gravity", because not only wasn't it proposed by Galilei, but he was actually confused by it, and it was one of the main reasons his heliocentric theories were rejected. It was theorized since classical antiquity that it's the property of all non-celestial objects that they strive to move towards the center of the Universe (which is the center of the spherical Earth). Without the Earth being in the center, this wouldn't work, so before Newton, heliocentrism was just a fringe theory which raised more questions than it answered. $\endgroup$
    – vsz
    Apr 14 '20 at 12:38
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Of course it does work on earth. Just consider these facts:

  • General Relativity also describes/encompasses 'standard' gravity. Just in a slightly over-complicated way.
  • Things 'thrown in the air' as well as satellites follow exactly the 'straight' lines (aka geodesics) of general relativity.
  • The GPS system needs to account for general relativistic effects in order to achieve its precision.
  • There have been experiments of transporting clocks around in planes that exactly confirmed general relativity.

Sure, it is not as dramatic as displayed in documentaries, but often in science the spectacular is hidden in the details.

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    $\begingroup$ Exactly. And Elephants do bend space around them, making sand grains curve around them. The effect is just way too small to be noticed! $\endgroup$ Apr 13 '20 at 10:58
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    $\begingroup$ Well, just imagine a commedian without any scientific background speaking that exact line... The statement is sound and true, but I guess it can ring like satire to an ear not accustomed to these matters. That's why I added the disclaimer. $\endgroup$ Apr 13 '20 at 15:02
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    $\begingroup$ I call this the "Pig Pen Effect" after the Peanuts character. ;) $\endgroup$
    – Engineer
    Apr 13 '20 at 20:00
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    $\begingroup$ @LawnmowerMan Well, I guess you could perform the Cavendish Experiment (en.wikipedia.org/wiki/Cavendish_experiment) with two dead elephants. The results would not be as precise due to the different weight of the two elephants, the lower density of the elephants and thus their greater distance to the test masses and other effects like the production of gasses by the rotting carcasses, etc. pp. I prefer the experiment to be done with plain old lead balls, not only for the scientific value... $\endgroup$ Apr 13 '20 at 21:12
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    $\begingroup$ @cmaster-reinstatemonica I bet everyone here is just dying to see that grant proposal... :-D $\endgroup$
    – SusanW
    Apr 14 '20 at 12:45
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It works pretty much. We just happen to live in an environment where Newtonian approximation of the GR is good enough.

We use the Newtonian approximation

  1. because of the tradition,
  2. because it is much, much less of a math hassle, and
  3. because the results of the approximation are of acceptable accuracy

As for why elephants don't cause object to orbit around them:

first, see https://en.wikipedia.org/wiki/Hill_sphere .

For short: the elephants in the general case are in the Earth's Hill sphere and it is the Earth here that distorts the spacetime the most. If you want an elephant-sized object to have its own satelites, you have to put it somewhere far away from Earth, so its Hill sphere to be larger than the object itself.

second, the air. Air is interacting with everything that tries to orbit the Earth in rather strong manner. Because of the air, every Keplerian orbit becomes a ballistic curve and the object goes to the Earth surface where the interaction is even stronger than in the air.

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Instead of spacetime curvature, it is better to think of a different system of coordinates.

For example, if someone is at 1 km from the North Pole, all the land (or ice) until the horizon in all directions (a radius about 5 km) can be approximately by a flat surface.

Even being flat, during any path in straight line, except if it is radial to the pole, the route is not constant, if by route we understand a given direction ($221^\circ$ for example, when $0^\circ$ is direction North. It is a consequence of using polar coordinates. But when the calculations are corrected for the effect of that curvilinear coordinates (covariant derivative) they show a constant straight velocity.

In a similar way, in our daily experience of a constant $g$ (approximately flat spacetime), any falling objects has a non uniform velocity in our coordinates of space and time (and also according to our senses in this case). But it also follows a straight constant velocity, in the meaning that the covariant derivative of the velocity is zero.

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  • $\begingroup$ Could you elaborate on the notation you're using where $0^0$ represents north? Is this a way of encoding latitude and longitude, or something like that? $\endgroup$
    – David Z
    Apr 13 '20 at 5:16
  • $\begingroup$ @DavidZ: In the standard notation for representing orientation, 0° is due north, 90° is due east, 180° is due south, 270° is due west. (See https://en.wikipedia.org/wiki/Azimuth#Navigation.) This answer is using a superscript-zero instead of a degree mark, but I highly doubt that that's intended to denote something different. $\endgroup$
    – ruakh
    Apr 13 '20 at 5:57
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    $\begingroup$ @ruakh Oh, the superscript $0$ is precisely the part I was asking about, not the standard way of indicating azimuth (which I'm quite familiar with). TBH it didn't occur to me that it might have been intended as a degree mark. I'll edit, but Claudio, if that's not what you meant, please revert my edit or make another one to clarify what you mean. $\endgroup$
    – David Z
    Apr 13 '20 at 6:45
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Or simply, how does spacetime curvature work inside a planet?

The curvature of spacetime is very much observable on Earth. Indeed, were spacetime not curved, then undisturbed objects would follow straight lines in spacetime, in other words, they would move with constant velocities; but objects released above the surface of Earth very visibly deviate from that, exhibiting acceleration towards the planet.

As for phenomena on Earth that can be explained by GR but not by Newtonian gravity, we can now make such tremendously accurate clocks that we can use gravitational time dilation to use them as glorified elevation measuring devices.

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  • $\begingroup$ The reporting in the link is confused about what is "first" in the 2018 paper it summarizes; see this from 2005. $\endgroup$
    – rob
    Apr 14 '20 at 9:08
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Elephants do have stress-energy and do create spacetime curvature.

Here on Earth on the small scales, the other forces (EM, weak, strong) dominate over gravity.

Still, you could ask, what would happen if you put an elephant into space (zero gravity) and put dust around it. Will the dust be gravitationally attracted to the elephant? Yes it will. This is how originally celestial bodies started to form from dust.

In astrophysics, accretion is the accumulation of particles into a massive object by gravitationally attracting more matter, typically gaseous matter, in an accretion disk.[1][2] Most astronomical objects, such as galaxies, stars, and planets, are formed by accretion processes.

https://en.wikipedia.org/wiki/Accretion_(astrophysics)

Now why do we not see the same effect here on Earth of an elephant having visible gravitational attraction on dust around it? It is because here on Earth, the spacetime curvature is dominated by the Earth's gravitational field, and it points towards the center of Earth. Dust moves (disregarding atmosphere) towards the center of Earth.

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  • $\begingroup$ This should be the accepted answer. Árpád explains in a very practical way, how GR is applicable everywhere and answers the OP's concern about the elephants. $\endgroup$
    – Timm
    Apr 16 '20 at 0:25
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The effects at the human scale is normally too small to be measured but it does happen.

Time dilation has been measured by using two accurate clocks one installed at ground level and one installed on a tower (see: Hafele–Keating_experiment). So that is one way general relativity affects real things at near-human scale on earth. Based on the experiment we can be confident that people at the top of skyscrapers experience time differently than people on the street.

More recently researchers have managed to use even more accurate clocks to measure time dilation by moving the clock a few centimeters vertically. (Also see: https://arstechnica.com/science/2020/04/portable-clock-provides-new-ruler-for-measuring-the-earth/)

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