If gravity is curvature of space, and everything is traveling in a straight line in the curved space, then why do two objects traveling with different speed in same direction from same point take different trajectory?
Q2) Curvature explains orbiting satellites, but then how does simple falling of objects is described?
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The curving of space is not just in space, but also time. Better combined together into _spacetime_. Lets explore why this answers your question:
All things are moving both in space and time. Even more, within this setup (and only then the curvature truly makes sense), everything is actually moving _exactly at the speed of light_! This may sound very weird at first, but normally almost all your speed is forward in time! Only if you get really _really_ fast in space can we notice that it now automatically moves slower in time: you age slower, things around you seem to speed up. Your temporal speed gets reduced.
Only with this realization will the weirdness of general relativity really make sense. Gravity is bending spacetime, both aspects of it. However, you are completely correct that two things with the same starting position and direction should travel the same path! Yet this seems to contradict observations, even basic ball throws, more so orbital mechanics.
The resolution of this paradox lies within the above: spacetime, and that everything always moves at the speed of light. When two objects start in the same direction but at different speeds, the one faster in space is actually slower in time! Put differently, their directions in spacetime are actually not the same; one moves faster “sideways” (in space), while the other moves quicker “forwards” (in time). And as a result of this, their resulting paths throughout bent spacetime will differ.
Edit: for the experts: yes I am aware that the sign for time is different; the above has been written in such a way to still be technically correct while obviously not dabbling in metrics and manifolds.
Think about putting a golf ball across the side of a hill. The harder you hit the ball, the less its path will seem to curve. The slope is “pulling” on the ball equally hard no matter what, but the faster the ball is going, the less time the slope has to “pull” on it before it gets to the other side.
Ok, first off, it’s true that objects in spacetime not under external forces follow “shortest paths” (called *geodesics*) through spacetime (I don’t think it’s helpful to think of this as a ‘straight line’). That said:
– These geodesics aren’t *unique*. A geodesic is a *local* minimum, meaning every *small* perturbation from that path is longer. There can be multiple geodesics connecting two points, as long as both are locally minimize the path.
– Even if objects follow “straight lines”, if they end up at different points they follow *different straight lines*. Even if they end up at the same point they can follow different “straight lines”, but if they end up at different points they certainly follow different paths.
It isn’t just space that curves, but *space-time*.
Objects with mass create a time gradient around themselves. The closer you get to an object, the slower time runs. Objects larger than point size experience a “time shear”, meaning part of the object closest to another moves through time slightly slower, and part of the object farthest away moves slightly faster. The slower moving portion essentially drags the faster moving portion through space, bending its trajectory, creating the illusion of acceleration from a particular reference frame.
This video explains it:
The force of gravity you feel from the ground below is essentially Earth’s matter constantly pushing you away from your inertial path.
Imagine 2 sailboats at the equator facing north, separated by a mile, let’s say. The boats start moving north at the same speed. From their perspective they are perfectly parallel to each other. Despite this, they will appear to be getting closer and closer together until, by the time they reach the North Pole, they will touch.
They may not understand why, if they were moving parallel, their boats would be drawn closer together and eventually meet. They may say that a “force” has been pulling them together.
Of course, we know that there is no force. It’s just the curvature of the surface of the earth that brings them together. They can’t perceive that curvature so they call it a force. They’re moving in a straight line on a curved surface, so it’s actually not straight.
Gravity is just like that. It’s a curvature in a higher dimension that we can’t perceive, so we call it a force.
Imagine 2 apples hurtling through space parallel to each other, separated by some distance. As time passes, they will get closer until they eventually touch. It appears to us that some “force” has brought them together, but it’s actually a curvature in a higher dimension that we can’t perceive that brings them together.
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