The Curvature Of The Geodesic In An SFL-Informed Ontology Of The General Theory Of Relativity

ChRIS said:

In an interpretation of general relativity in which gravity is the contraction of space intervals in the direction of the centre of mass, can you explain why this results in a curved geodesic for a small body passing a much larger body. Keep in mind, in this model spacetime is not curved. Only the geodesic path through space is curved, not space itself.

ChatGPT said:

Here’s how this works:

1. Space Contraction (No Curved Spacetime): In your interpretation, spacetime itself remains flat—there’s no warping or bending of spacetime as in the standard interpretation of general relativity. Instead, gravity is understood as a contraction of space intervals in the direction of the central mass. This contraction affects how distances are measured along different directions in space, but spacetime itself doesn’t undergo curvature.

2. Geodesics as Curved Paths Through Space: Geodesics represent the straightest possible paths through space in the absence of forces. However, since space itself is contracted in the direction of the central mass, the straightest path for the small body, in the usual sense, will appear curved when observed from a distant point. This is because the space through which the small body travels is not uniform; in the direction toward the large mass, space intervals are contracted, leading to a trajectory that looks curved from an external observer’s perspective.

3. Curved Path Resulting from Space Contraction: When a small body passes near a larger body, the distances it traverses along the direction of the central mass’s gravity (the line pointing toward the mass) are contracted more than those in the perpendicular direction. The small body might think it’s following a straight line in its local space, but the contraction of space intervals means the distance it covers in the direction toward the mass is shorter than it would be if the central mass weren’t present. This causes the path of the small body to appear curved as it moves through the contracted space, even though spacetime itself isn’t curved.

4. Visualising the Geodesic: Think of the geodesic as a "straight" path through space, but space is not uniform. As the small body moves through space, the contraction in space intervals (specifically along the radial direction) alters the straight-line path, causing it to deviate from what would otherwise be a straight trajectory. This deviation gives the appearance of a curved geodesic in space, despite spacetime being flat.

In summary, the curvature of the geodesic in this model is a result of the differential contraction of space intervals along the path of the small body, not from curvature in spacetime. The small body moves through a space that is contracted in the direction of the larger mass, and this results in a curved path, even though spacetime itself remains flat.