So I’m familiar with mass affecting space time, notably the grided flat sheet representing it. All objects with mass bend this sheet. The effects of massive objects are represented by the object creating wells, with the well being deeper and steeper the more massive the object is. I know this isn’t the actual explanation, but it helps visually.
I know gravity is not a force, and the reason mass “attracts” other objects is because those objects are going in the direction of the distorted space time created by the “attracting” object.
So why does mass cause a greater acceleration the more massive the attracting object is? What difference would it make if the gravity well was “shallow” or “deep”? Gravity isn’t a force, so what is actually pulling an object faster into the well depending on its depth?
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Technically it’s not just mass that curves spacetime, but it’s “stress-energy” which includes mass, energy, pressure, etc.
The ten Einstein Field Equations (EFE) are a set of equations created to model gravity and those equations create an equality between the Einstein tensor (which is what describes spacetime curvature) and stress-energy. This equality would suggest it’s more accurate to say that mass doesn’t cause spacetime to curve but that mass **literally is** spacetime curvature.
One thing worth pointing out is that the flat sheet bending around massive objects is a stylized, learning tool. Before General Relativity we imagined space as an infinitely large box that objects were oriented in and time as simply a snapshot of the complete orientation of each object in the box. General Relativity revealed that there is no such box, no “background” that objects reside in. Spacetime is our attempt to continue having a background and even then that background is essentially an object requiring careful examination.
Another thing to consider is the idea of curvature. If you draw a circle on a sheet of paper, you might argue that the circle has curvature. In geometry we’d say the circle has “extrinsic” curvature, but the circle has no “intrinsic” curvature. General Relativity uses intrinsic curvature. Extrinsic curvature would suggest that space is curved relative to the space it is embedded in, which we just explained is no longer how we view the universe. Intrinsic curvature can be measured without requiring an external space to refer to.
This is where the “curved sheet” comes in: the path a particle follows through spacetime is actually a straight line, it’s just curved when you map it to a subspace like a sheet. This line is actually called a **geodesic**. In Euclidean geometry, the geometry you’re most familiar with, the shortest distance between two points is called a straight line, which is a geodesic. In this case the geodesic is minimizing the path length to travel from point A to point B.
In General Relativity the geometry now has to combine space and time into spacetime. This effect causes the sign in the mathematics to reverse and instead the geodesic follows the path that **maximizes** the spacetime length of the path. The effect of maximization physically manifests as the amount of time measured by a particle being maximized. Note that this is just Newton’s first law of motion occurring in a curved spacetime now. An object experiencing no external forces will follow a straight-line path at a constant velocity. Imagine an apple and Earth both following a straight-line path (through a curved spacetime) that intersect in a future state.
All of this is to say that geodesics mathematically manifest in a way such that the world-lines of objects with mass “accelerate” towards each other. Remember also that the entirety of spacetime is curving, not the spatial path of objects. An object “at rest” is still in motion through time and an object moving through the “time direction” will gradually move towards the “space direction” in a curved spacetime.
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