The space bends
When space bends, what lines are bent?
When people talk about the curvature of space they are usually talking about two related ideas: 1) the curvature of the world lines in 3 + 1 space-time or 2) the curvature of the light paths in 3 space. These ideas are related when light behaves as a high speed, low mass boundary.
1) Consider a 2D surface embedded in flat 3D space. There is a tangent plane at every point on the surface. If the surface is curved, moving the tangent plane from one point to another will change the angles it makes with respect to the fixed three-dimensional axes.
Now consider a particle moving at a constant unit speed along this surface. We can project its velocity vector onto the fixed three-dimensional axes. At any point along the curve, the components of the velocity change, but the overall magnitude of the velocity remains the same.
To make the connection with spacetime we let one dimension be time, the other two are space, and our constant unit velocity is the velocity of the correct time of the particle. The curve followed by the particle is its world line, and the ratio of its spatial components to its time component is the observed velocity of the particle. If we look at a particle moving in a straight line in flat space, a force that accelerates it causes its world line to curve towards the spatial plane, and a force that causes it to slow down causes that the world line curves in the direction of the time axis.
2) To describe a 3D space that is not flat, I like to think of sponges. A flat room is represented by a homogeneous sponge. When we place a mass in a homogeneous sponge, it pulls the sponge towards it with a force that depends on the distance. The effect is that the sponge becomes thicker and denser as you get closer to the crowd.
A ray of light seeks the straightest path with maximum spatial component and constant speed that it can find. While particles can slow down / accelerate when the density of the space sponge changes, the light beam does not have this option so it is deflected to optimize its path.
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