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In general relativity something in free fall, that appear to accelerate towards the earth, is actually not accelerating at all but moving along a geodesic so why does it appear that it is accelerating to us relative to the earth?

Is it because the earths surface is accelerating up into it?

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    $\begingroup$ Related: physics.stackexchange.com/q/102910/2451 , physics.stackexchange.com/q/3009/2451 and links therein. $\endgroup$
    – Qmechanic
    Dec 17, 2021 at 19:32
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    $\begingroup$ The you tube channel ScienceClique youtube.com/ScienceClicEN covers quite intuitively all these concepts. The have a series on GR, that you might like and find an answer to. $\endgroup$ Dec 18, 2021 at 0:37
  • $\begingroup$ the important keyword is "geodesic" so the trick is you apply vectors on every crook and cranny, atom and molecules of the object in motion, unfortunately spacetime is curved just imagine those vectors converge and become shorter as it moves inside this distorted geometry of spacetime. So there you go Earth bends spacetime and object moving in geodesic within this distortion of spacetime appears to accelerate! $\endgroup$
    – user6760
    Dec 18, 2021 at 4:03
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    $\begingroup$ youtu.be/XRr1kaXKBsU you might want to watch this Veritasium video $\endgroup$ Dec 21, 2021 at 2:49

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The object accelerates downward as measured by the coordinates expressing time and height that we use. So, in a certain pratical way, the acceleration is real, because we measure it.
One analogy is to associate time to longitude and height as latitude in a region very close to the North pole, let's say 1 km around it. Being at rest at the earth surface is equivalent to travel along a parallel (so keeping the same latitude). But travelling in a straight line, making a chord between 2 points of this circle of 1 km radius, requires going to higher latitudes and later on return to the initial one. If the traveller following the straight line relies on the coordinates, the ratio $$\frac{\Delta Lat.}{\Delta Long.}$$ is not constant, so the movement is 'accelerated'. It can be compared to a stone that we throw upwards. It also goes up until reach a maximum point, and comes back afterward.

We can correct it in the analogy by making a rectangle, using the chord as an edge, and deploying cartesian coordinates for time and space instead of longitudes and latitudes. In this case all works fine, and straight lines are represented as constant ratios between coordinates.

In the real world, it is like being in the stone frame. All other objects in free fall will be travelling at constant speeds for that frame. But it works for small time intervals and small height.

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Is it because the earths surface is accelerating up into it?

It's more like it's just different directions in spacetime. The shortest path between today and tomorrow is through the center of the earth.

In 4-space everything, even objects "at rest" are moving. You can see this when you look at your watch. Sit still all you want, it keeps ticking. The velocity through 4-space is the speed of light, or if you prefer, one second per second. The key to this is that that velocity is invariant: the speed of light is the same in all frames.

So the situation is much like a ball at the top of a hill in terms of conservation of energy. As the ball rolls down the hill, you lose potential energy and it turns into kinetic. In this case, as you "roll down the gravity well" you are losing speed along the time axis (which is being "bent" from the earth's perspectiv) which means you have to gain it in the other axes in order to conserve the original velocity.

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  • $\begingroup$ "The shortest path between today and tomorrow is through the center of the earth." I would suggest modifying or removing this sentence. Very few free-fall trajectories go through the center of mass. $\endgroup$
    – g s
    Dec 17, 2021 at 23:20
  • $\begingroup$ "The velocity through 4-space"- i could be incorrect but this idea seems wrong to me. If our time passage speed means our speed through a time dimension which is same for all things then, a person standing on earth decompose into smaller and smaller parts and into different time lines. $\endgroup$
    – user316791
    Dec 23, 2021 at 5:25
  • $\begingroup$ You can take an analogy:- " Assume that there is a straight road representing time-line. There are two cars on the road, one moving faster than second one. You will see that if both cars started at same time at same position, the front of one car with high speed through time will move more. And in few time that car would be a large distance apart from slower car. Also the person in one car cannot see its back at any time" Means if two things are having different passage of time then, they both will get lost in time-line in no long time. Same with you head and foot. $\endgroup$
    – user316791
    Dec 23, 2021 at 5:25
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Is it because the earths surface is accelerating up into it?

I am not a physicist. I don't know whether it is proper to say that the Earth's surface is "accelerating up," but when you stand on the Earth, the force that you feel pressing against your feet is due to the Earth's surface preventing you from following a geodesic (i.e., preventing you from freely falling.)

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  • $\begingroup$ I actully think that the idea of earth accelerating upward is pretty dumb idea. $\endgroup$
    – user316791
    Dec 17, 2021 at 22:26
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    $\begingroup$ @ArsenalCreation, But if you wake up, locked in a windowless room on board my space ship, and you can feel the floor pushing up on your feet, there is literally no way for you to tell whether you feel that force because the the ship is resting on the Earth's surface, or whether you feel it because the ship is accelerating through space at 1G. $\endgroup$ Dec 18, 2021 at 0:17
  • $\begingroup$ But this is not the case with earth. If this idea is true then, then earth would be expanding outward with an acceleration. This idea may rise the question "what is the expansion acceleration of object dependent upon" is it its mass or volume. You know that there are isotopes of same size which means that this acceleration should not be dependent upon mass otherwise you would see one isotope becoming larger and larger as compared to other. This idea is completely wrong and i have also tested this idea and found that this was wrong idea $\endgroup$
    – user316791
    Dec 18, 2021 at 10:47
  • $\begingroup$ @ArsenalCreation youtu.be/XRr1kaXKBsU you might want to see this video where he explains the Earth expanding part at 9:55 timestamp $\endgroup$ Dec 21, 2021 at 2:46
  • $\begingroup$ I actually don't understand that part, specially the equation. He should have explained about that in more detail. But does it mean that earth is expanding in space with an acceleration? If yes then, what is this acceleration dependent upon? Newtons says force is dependent upon mass so you would like to say that expansion's acceleration must be dependent upon mass. If no then what is it dependent upon? If yes then i already gave a situation of isotopes of an element having same size and different mass at a given point of time which seems to violate the thing. As much i think $\endgroup$
    – user316791
    Dec 21, 2021 at 6:48
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The answer to your question is the four velocity and the fact that we happen to live in a universe where the four velocity vector's magnitude has to stay constant.

Even objects "at rest" (in a given reference frame) are actually moving through spacetime, because spacetime is not just space, but also time: apple is "getting older" - moving through time. The "velocity" through spacetime is called a four-velocity and it is always equal to the speed of light. Spacetime in gravitation field is curved, so the time axis (in simple terms) is no longer orthogonal to the space axes. The apple moving first only in the time direction (i.e. at rest in space) starts accelerating in space thanks to the curvature (the "mixing" of the space and time axes) - the velocity in time becomes velocity in space. The acceleration happens because the time flows slower when the gravitational potential is decreasing. Apple is moving deeper into the graviational field, thus its velocity in the "time direction" is changing (as time gets slower and slower). The four-velocity is conserved (always equal to the speed of light), so the object must accelerate in space. This acceleration has the direction of decreasing gravitational gradient. Edit - based on the comments I decided to clarify what the four-velocity is: 4-velocity is a four-vector, i.e. a vector with 4 components. The first component is the "speed through time" (how much of the coordinate time elapses per 1 unit of proper time). The remaining 3 components are the classical velocity vector (speed in the 3 spatial directions). $$ U=\left(c\frac{dt}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau}\right) $$\

If in your example, you put a object initially at rest (relative to Earth) into Earth's gravitational field, then General Relativity tells us that the Earth's gravitational field will have an effect on the object, it will slow it down in the temporal dimension (GR time dilation). Now this means that the object's four velocity's temporal component will change. Remember, the magnitude has to stay constant, so the spatial components will have to compensate (change), meaning, that the object will start moving towards the center of mass of the Earth.

You could say that the objects are trying to reach a balance between the velocities in the different dimensions, but rather, one of the ultimate messages of General Relativity is that an objects velocity is not independent in the different dimensions, the object's velocity in the temporal dimension affects the velocity in the spatial dimensions and vica versa.

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