The concept of curved or warped space is simple to understand when viewed mathematically, although the concept is counter intuitive.
In pure 3D geometry, the Pythagorean is valid in a Euclidean or flat space.
- DeltaDistance3D[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaZ[sup]2[/sup]
DeltaDistance2D[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup]
In a curved or warped space, the above formulae for distance are incorrect & more complex formulae must be used. In the above, DeltaX, DeltaY, & DeltaZ are the differences in the coordinates for two points. DeltaDistance is the distance between the points.
The above concept is not too difficult to visualize for curved 2D spaces like the surface of a sphere. One consequence of the failure of the Pythagorean theorem on the surface of a sphere is that the interior angles of a triangle do not add up to exactly 180 degrees. The concept is far more difficult for our intuition when dealing with a curved 3D space, but the mathematics is not much more difficult than for curved 2D spaces.
A convenient method of expressing the laws of physics introduces the concept of an event which occurs at (X, Y, Z, T) & allows the laws of physics to be analogous to the geometry of a 4D space. This is a convenience, not a necessity.
For a flat 4D space time continuum, the interval between events is given by the following formula.
- DeltaDistance[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaZ[sup]2[/sup] - DeltaT[sup]2[/sup]
The above assumes that the units are such that the speed of light is one. Otherwise, the formula is
- DeltaDistance[sup]2[/sup] = DeltaX[sup]2[/sup] + DeltaY[sup]2[/sup] + DeltaZ[sup]2[/sup] - (c*DeltaT)[sup]2[/sup]
Special Relativity uses the above for the distance between events. It is assumed that gravity does not exist or that it is too weak to have an effect. Special Relativity applies to events occurring in the void between galaxies.
General Relativity takes gravity into consideration & similarly expresses the laws of physics using the concept of an event occurring at (X, Y, Z, T). The formula for the distance between events is more complex, indicating that the laws of physics with gravity included are analogous to the geometry of a curved Space-Time continuum.
Note that using mathematics which views the laws of physics as analogous to the geometry of a curved 4D space-time continuum is very convenient, but not necessary.
Saying that gravity warps space is actually jargon relating to the mathematics used. You do not have to believe in curved space or that gravity warps space to deal with the laws of physics. It is merely a convenient model.