Spacetime diagram

Here is a handy diagram of spacetime, with a brief explanation of how it works.

This is a diagram of the potential energy field, better known as spacetime.  At the center is a stationary particle of mass 8, along with curves showing its effects upon spacetime.  (Don't ask 8 what.  It's just 8 units of energy.  Doesn't matter.)  The horizontal axis is space (which is equivalent to time, because if it weren't, nothing would ever move or change).  The vertical axis is potential energy, which is equivalent to time.  Our particle is using up 8 units of energy, so it depresses the potential energy field by that much.  This is what work is.  (The total work done by the particle is mass plus kinetic energy.)  (In this graph, the top of the potential energy field is zero, where proper time = 1.  The bottom is somewhere, way, way off the chart, where proper time = 0.)

The red curve is the position of an approaching particle.  It's abstract, but the important part is that it can't be negative, because that would be inside our test particle.  So the effects of our particle begin at distance 1 from the center.  Don't ask what the units are.  Doesn't matter for the purpose of this explanation.

The purple curve is the force of attraction (AKA gravity) to another particle.  Any other particle.  Other particles are attracted to our test particle by empty space pushing them down the gradient.  Once again, just because the math shows the curve approaching zero in the center, doesn't mean it actually does.  The particle is a sort of discontinuity with radius 1.  Why is this curve following the inverse square law?  Because energy is conserved and the field is elastic.  The area above this curve is gravitational potential energy.  The field elasticity allows changes to propagate at a finite rate - the speed of light, c.

The green curve is the velocity of a particle attracted by gravity to our particle.  This curve is hyperbolic spacetime.  Just because the math shows it approaching zero, doesn't mean it really does.  The particle is absorbing 8 units of potential energy.  The curve crosses this mark at 1 unit from the center.  That's where the curvature ends.  Or, rather, where it turns into a spheroid of radius 1, because space is three dimensional, while our graph is not.

The slope of the green curve is really interesting.  The horizontal part is proper time and space contraction.  The vertical part is speed (which goes to the speed of light when the slope is vertical, and zero when the slope is horizontal).  The hypotenuse is a constant, c, the speed of light.  The trick here is that this green curve is the path that particles follow as they move.  The speed of light (propagation of change) is constant along this curve.  Not along the horizontal axis.  This has profound and weird effects, because what we perceive is three dimensional space.  We can't "see" time, only space.  So what we see is the vertical projection along the horizontal axis of what is really happening along the curve.  A 3D projection of the true 4D reality.  (Want to know what a tesseract looks like?  Look at a cube.  Now look at it again.  Tesseract!)

How does the particle itself move?  It has its own slope across its diameter.  That's why it is a spheroid, not a sphere.  It squishes along the direction of motion.  (Geometrically, it can be described as an ellipsoid, usually a spheroid.)  This is a locally hidden variable, but one we can easily observe the effects of.  (Bell's theorem is bunk and hokum.  It proves nothing, because it's a straw man argument.)

Imagine a small object near the edge of the graph.  It is forced towards the center by the gradient, picking up speed and energy along the way.  But as it gets closer to the center, the slope of the green curve increases, shrinking the apparent size of the object along the horizontal axis.  This is what we perceive as the time dilation and length contraction of general relativity.   The object's own contraction in the direction of motion due to its increasing speed is special relativity, and is caused by its own gradient.

Notice the green and purple curves cross at radius 1 (and at the given mass, in this case 8). No matter what value you pick for the mass, they always cross at radius 1, the same place the red position curve drops to zero.  This is not a coincidence.  This is what prevents singularities and infinities from happening in the real world.

• Space is flat (triangles add up to 180 degrees) and three dimensional.
• Spacetime is curved (hyperbolic) and four dimensional.
• Motion is along curves in spacetime.
• The propagation of change along the curve is finite and constant (the speed of light).
• What we perceive as motion is a 3D projection of the 4D reality.
• Particles are tiny, spheroidal discontinuities.
• The particle is work done on the potential energy field.
• The area above the force curve, outside the particle, is free (available) energy.
• Gravity isn't really attraction, it's empty space pushing down along the gradient.
• Every particle believes local spacetime is flat.  They're all wrong.
• Every particle has properties hidden inside the discontinuity, like velocity.
• Special relativity is caused by the slope of a particle's own velocity gradient.
• General relativity is caused by the slope of the spacetime gradient across a particle.

There are no infinities.  There are no singularities.  There are no contradictions.