# Curved spacetime GR vs Newtonian physics

In GR Mass is said to curve spacetime which is the explanation for Gravity versus Newtonian physics where it is explained as all matter attracts each other. My question is how does curved spacetime cause objects traveling at different velocities to follow different paths? If gravity was only due to spacetime being curved shouldn't objects follow the same path they are own regardless of their velocity?

In Newtonian physics it makes sense that objects with higher velocity with regard to a gravitating mass would be less effected because they spend less time near it as they pass by and so less acceleration is done to them. the object would follow a curved path dependent on it velocity with a flatter curve the faster it travels.

So, the only explanation I could come up with in GR would be that curvature of spacetime is relative velocity dependent meaning that the faster you are traveling relative to another object the flatter spacetime is due to gravity from that object. Is this what GR actually says? or is it explained another way?

DaveC426913
Gold Member
My question is how does curved spacetime cause objects traveling at different velocities to follow different paths? If gravity was only due to spacetime being curved shouldn't objects follow the same path they are own regardless of their velocity?
Ever seen a golfer hit a putt too hard? It rounds the hole and keeps on going on a deflected trajectory.
The faster the ball, the less time it spends in the higher curved zone, the less deflected it gets.

@DaveC426913 The issue with examples such as the one you give assumes the gravitational force of the golf ball and the earth on the Z-axis. It is this force that causes different velocity balls to follow different paths. The explanation in GR (as i understand it) is that there is no such force, it is only an illusion. objects follow the paths they do only due to curved spacetime, nothing else.

DaveC426913
Gold Member
@DaveC426913 The issue with examples such as the one you give assumes the gravitational force of the golf ball and the earth on the Z-axis. It is this force that causes different velocity balls to follow different paths. The explanation in GR (as i understand it) is that there is no such force, it is only an illusion. objects follow the paths they do only due to curved spacetime, nothing else.
Yes, it is a ham-fisted analogy.

I was simply put out there to show how velocity is a factor in the deflection of a trajectory through a curved environment.

Still why do you think it doesn't apply? It doesn't require a force to work. If you were to draw line segments on a curved surface - with their length representing velocity - lines with longer segment show less curve.

Nugatory
Mentor
If gravity was only due to spacetime being curved shouldn't objects follow the same path they are own regardless of their velocity?
It would work that way if it were just space being curved, instead of spacetime - the surface of the earth is curved in space but not time, and your straightline path along that surface is the same regardless of speed.

But it's spacetime that is curved, and that's different. First, work through this video (by our own member @A.T.)
to see how curved spacetime causes a dropped object to accelerate towards the surface of the earth.

Something moving relative to the apple will follow a different path through spacetime and will "feel" a different curvature;. It's like the difference between walking straight up a steep hill and walking up at an angle; the amount of vertical distance covered per unit of sideways movement will be different. There's an explanation, unfortunately not as clear as A.T.'s video, here: http://curious.astro.cornell.edu/ph...nd-bullet-follow-different-paths-intermediate

There's also an excellent illustration in Misner, Thorne, and Wheeler's classic GR textbook, but I don't know if this illustration is available on line.

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• bhobba
DaveC426913
Gold Member
Nugatory brought the diagram to the table that I was going to refer to. :)

In GR Mass is said to curve spacetime which is the explanation for Gravity versus Newtonian physics where it is explained as all matter attracts each other. My question is how does curved spacetime cause objects traveling at different velocities to follow different paths? If gravity was only due to spacetime being curved shouldn't objects follow the same path they are own regardless of their velocity?

In Newtonian physics it makes sense that objects with higher velocity with regard to a gravitating mass would be less effected because they spend less time near it as they pass by and so less acceleration is done to them. the object would follow a curved path dependent on it velocity with a flatter curve the faster it travels.

So, the only explanation I could come up with in GR would be that curvature of spacetime is relative velocity dependent meaning that the faster you are traveling relative to another object the flatter spacetime is due to gravity from that object. Is this what GR actually says? or is it explained another way?
One cool thing I saw in Reflections on Relativity (a popular online book often posted here) is that Newton's model can be described as the time aspect of GR. I know this seems tangential, but if you think about it, it has some relevance.

Specifically, this section:

https://www.mathpages.com/rr/s8-05/8-05.htm

"In essence, the effect of Newtonian gravity can be explained in terms of the flow of time being slower near massive objects, and just as a refracted ray of light veers toward the medium in which light goes more slowly (and as a tank veers in the direction of the slower tread-track), objects progressing in time veer in the direction of slower proper time, causing them to accelerate toward massive objects."​

A.T.
My question is how does curved spacetime cause objects traveling at different velocities to follow different paths?
Paths in space, are different from paths in spacetime. The same path curvature in spacetime, can produce different path curvatures in the projection onto space. The curvature of the spatial path depends on the orientation of the spacetime path (initial velocity).

• Ibix
PeroK
Homework Helper
Gold Member
2020 Award
In GR Mass is said to curve spacetime which is the explanation for Gravity versus Newtonian physics where it is explained as all matter attracts each other. My question is how does curved spacetime cause objects traveling at different velocities to follow different paths? If gravity was only due to spacetime being curved shouldn't objects follow the same path they are own regardless of their velocity?

Curved spacetime doesn't mean that space has a specific shape that objects must follow: like being on a rollercoaster, where you are constrained by the shape of the track. A particle in curved spacetime is, in principle, free to move along any trajectory. There are no physical constraints as such.

However, in order to explain any motion, you need a law of motion to say what paths are possible and what are not. In flat spacetime, where we have SR, the allowed paths are the same as in Newtonian physics: straight lines at constant velocity.

The real question is what principle governs the laws of motion in Newtonian Physics, SR and GR.

Newtonian physics can be described by forces acting on particles. But, it has an equivalent formulation based on minimising the action integral of the Lagrangian. In simple terms, instead of forces being responsible for motion, nature is trying to minimise a certain quantity. Look up Lagrangian mechanics.

In SR, the equivalent Lagrangian principle leads to nature trying to maximise a certain quantity: the "proper" time that the particle experiences.

If you extend this principle to GR, that particles move so as the maximise their proper time, then that and the specific curvature of spacetime gives you the predicted motion for a particle.

Finally, you can in fact, reproduce Newton's theory of gravity by curved spacetime. It's not quite the same as the curved spacetime predicted by GR, but it's similar. This gives you, at least, three ways to look at Newton's Theory of Gravity:

1) Forces and "Newtonian" mechanics
2) Gravitational potential and "Lagrangian" mechanics.
3) Curved spacetime and an extension of Lagrangian Mechanics.

If someone had been really clever, they could have come up with 3) and the concept of curved spacetime before Einstein, just as another way of describing Newton's theory.

haushofer
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