gravity curved space geometry einstein general relativity

#Gravity curved space geometry einstein general relativity free#

So consider two nearby longitude lines, at the equator they are parallel but they intersect at the poles. Longitude lines are geodesics, so they qualify as straight lines in a curved space (a sphere). This is geometrically impossible in flat spacetime, but is possible in curved spacetime.Ĭonsider the surface of a sphere: the concept of straight lines generalized to what is called a geodesic and on a sphere the geodesics are great circles.

as they fall through holes drilled to the center of a planet). With tidal gravity two objects can initially be at rest (parallel) and have zero proper acceleration (straight lines) but still collide (e.g. However, in the presence of tidal gravity this no longer holds.

So, two parallel straight lines are objects that are at rest with respect to each other and whose accelerometers read 0. And straight lines in spacetime correspond to objects with accelerometers that read 0. Now, parallel lines in spacetime correspond to objects at rest with respect to each other. Proper acceleration is the physical acceleration directly measured by an accelerometer so this relationship is very convenient and has a direct experimental significance. The radius of curvature of this line is directly related to the proper acceleration with a sharper curve corresponding to a greater proper acceleration.

In contrast, an object which is subject to a force will accelerate, which is represented by a curved line in spacetime. In a spacetime diagram this is simply a straight line. In flat spacetime an object which is experiencing no forces will travel in a straight line at constant speed, as dictated by Newton’s first law.

#Gravity curved space geometry einstein general relativity free#

I still have a lot of fog about this part but I think I read somewhere that Einstein had some struggles when he needed to describe gravitational field as a mathematical object with 10 components, also what is its connection to coordinate free differential geometry? I want to know more about these points.įirst, we need to make sure that you understand how motion and acceleration without gravity works in pseudo-Euclidean (flat spacetime) geometry before we can explain why pseudo-Riemannian (curved spacetime) geometry is required to describe gravity. I'm looking for an explanation that starts with the Euclidean geometry/Cartesian coordinate description of gravitational field, and ends with differential geometry/metric tensor field/(something else ?) description. Furthermore you can treat gravity like a fictitious force in a free falling frame: two objects initially at rest with zero proper acceleration just follow their geodesics.īut the answer I have been looking for since the beginning is more quantitative than the above. In Einstein's picture, space is like a trampoline surface, when empty, it's flat, you put a ball on it, the ball curves the surface and the distortion is local and spreads at speed c.

There's also another issue with tidal gravity in a reference frameįree falling in gravitational field (in such frame, you can't transform gravity away if space is described by Euclidean geometry because if you could you'd have two objects initially at rest with zero proper acceleration moving towards each other, or, two parallel straight lines intersect).

There's an issue with action at distance (which seems like a huge problem after the postulate about speed of light was made).

I think I'm getting close to an understanding from piecing together various answers. I want to ask more specifically, how? Basically I was expecting answer that contrasts the old gravity (wasn't it just a vector field?) and the new gravity (something more complicated)? Einstein found the old geometry somehow insufficient to describe his new idea about gravity, I want to know more about this part.Įdit 2. So gravity is the reason that geometry of space can't be Euclidean (correction: this is not quite right, as explained by 2 users). How would you explain to someone the change that Einstein needed in geometry for his new ideas about gravity and spacetime, what did he seek but could not be described by pseudo-Euclidean geometry?Īssuming you're talking to a highschool student who only knows Euclidean geometry, Newtonian physics, Calculus and Cartesian analytic geometryĮdit.