What dimension does space-time curve in?

[alw34]

As I fall towards the surface(experiencing gravity) , do I still need to fire rockets to stay on the straight line perceived by myself or someone on the planet?

Just to be clear: you normally can 'perceive' straight line paths in spacetime by whether or not you are feeling acceleration...see the earlier post of about hanging apple on a stem versus free fall for one example. Here is a bit more...

A geodesic is a free fall worldline; A geodesic has zero [proper] 4-acceleration and zero curvature. You don't feel any acceleration.

When you don't 'feel' acceleration, you are in free fall and moving in a straight line [geodesic] in spacetime. This may appear curved, as already described, in space alone. [exactly as described in post #28] Analogous to an apple stem holding an apple in place, sitting in a chair in your home IS properly-accelerating. You feel it on your backside, and a test device called an accelerometer will register acceleration.

Feeling acceleration is the telltale sign you are NOT moving in a straight line in spacetime.

It's sometimes counter intuitive: going down the highway in a straight path through space,a 'flat highway' you still feel gravity on your backside,so you know you are not going in a straight line in spacetime. Then you come to a curve in the road, in space, and now you feel additional acceleration, that's how you know you are moving in a different curve in spacetime. And when you go up or down hills at constant speed you also feel different accelerations which tells you your path through spacetime is curved differently.

When would you be going in straight line in spacetime? When you run off a bridge and are in free fall, just like the Earth to Mars example in the prior post. Of course, it looks curved in space, a parabola is it??. Our puny senses can't detect the very small effect in time.

 

[pervect]

I will post an image of an "easy to visualize" trip on the globe that illustrates the point Odoruin was making. The trip follow four geodesic paths, each of which is at right angles to the other. All segments of the path has the same length, and starting and ending points of the journey are different. This is not possible on a plane.

To recap - the equivalent of straight lines on the globe are called geodesics, they represent the shortest route that you can take between two points on the globe without leaving the surface of the globe. On a spherical globe, these geodesic paths turn out to be great circles. I hope the concept of right angles is reasonably clear, rather than muddy up the exposition by going into details, I'll assume for now that it is. If questions arise, we can address that in a separate post.

Onto the picture of the journey. A short summary - the trip starts at A. Th four legs of the journey are A-B, B-C, C-A, and A-B. Thus the trip starts out at A, and winds up at B.

You start at point A, which is on the equator, at 90 degrees west longitude. You go north until you reach the North pole, at B. Next, you make a right angle turn, and go south down along the line of 0 degrees longitude, back to the equator. You make another right turn (the third), and head west along the equator. At this point you've taken only three legs of the journey, but you're already back at your starting point of your journey! To finish the trip, you make another right turn, and take the last and final leg of your journey, which goes back up to the North pole.

So, following the geodesic lines on the globe, here we have an example of a trip that takes 4 right angles, wherein all legs of the trip have the same length (1/4 of the circumference of the globe). The trip starts on the equator, and winds up on the north pole.

This then illustrates the difference between a curved spherical geometry and a flat plane geometry. On the spherical geometry you can start at a point, travel along what is essentially a squre (a quadrilateral with four right angles and equal sides), and wind up at a spot different than where you started. This is not possible on a plane geometry, and it illustrates how you can determine the existence of curvature via geometrical meansm without ever leaving the curved surface, simply by drawing geodesic lines which can be understood (slightly oversimplified) as the shortest distance between two points.

 

[A.T.]

When you don't 'feel' acceleration, you are in free fall and moving in a straight line [geodesic] in spacetime.

Just want to point out for the OP that "feel acceleration" means that an inertia based accelerometer, that you hold measures non-zero accelration. This is for example the case when you sit at your desk.