Curvature of Spacetime

How Spacetime Is Commonly Represented today

To illustrate spacetime curvature, scientists often use the metaphor of a sphere distorting a flexible grid. The internet images below show how common this representation is.

These metaphors are instructive because they suggest that spacetime deformation is produced by volume, not by mass. Indeed, mass is not explicitly represented in these illustrations.

If we think strictly in terms of mass, as academic physics does, these figures do not explain the origin of spacetime curvature. This leads to a central question: how can a mass distort spacetime?

Spacetime Curvature: A Genuine Mystery

Imagine a glass of water filled to the brim. The water represents spacetime. Now drop a ball into the glass. What causes the water to overflow — the ball’s volume or its mass? Clearly, it is the volume.

According to General Relativity, spacetime distortion is caused by mass, not volume, which creates a paradox. Hence the question:

• Is spacetime curved by volume? This idea aligns with intuition, as illustrated by the ball in water analogy, but it contradicts Einstein’s theory.
• Is spacetime curved by mass? Conversely, the view that spacetime is curved by mass is fully consistent with General Relativity, yet it defies intuition. Claiming that the ball’s mass causes the water to overflow is counterintuitive. Regardless of the material — wood, iron, or anything else — the overflow is due to the ball’s volume, not its mass.

Resolving the Paradox

The key is that spacetime is distorted by "Mass Volumes", as defined in the previous chapter. There is no contradiction because:

• Mass Volumes, or "volumes with mass": These are the volumes that curve the environment in which they are immersed.
• Mass: Since Mass is present within mass-volumes, this aligns perfectly with General Relativity.

Therefore, it is the mass-volumes that curve spacetime. This reconciles Einstein’s description of mass with the intuitive notion of volume.

Note: Empty volumes are transparent to spacetime because they are void. They are not considered in this section.

Minkowski vs. Einstein Spacetime

The differences are:

• Fig. A: Minkowski spacetime is empty and linear because it contains no mass-volumes.
• Fig. B: A corpuscle, such as an electron or nucleon, introduces mass-volumes, transforming Minkowski spacetime into Einsteinian (Riemannian) spacetime. Since spacetime cannot penetrate mass-volumes, it must contour them, producing curvature — analogous to water displaced by a ball in a full glass.

The Apparent Volumes Mix-up

Apparent volumes differ from mass-volumes by their mixture of mass and vacuum. Their spacetime curvature depends on this ratio, which directly affects quantitative predictions.

To understand curvature, one must ask: what causes it? Not mass alone, but mass-volumes — like a ball displacing water in a full glass.

It is therefore essential to use the appropriate volume type in calculations: empty (Fig. A), mass (Fig. B), apparent (Fig. C), or hermetic (not shown).

Apparent Volumes in Spacetime

Spacetime curvature in apparent volumes depends on their closed volume content. The ball example illustrates this (above figure):

• Fig. A: Minkowski spacetime with only empty volumes — no curvature.
• Fig. B: A mass-volume ball in water raises the level by its full volume.
• Fig. C: A perforated ball becomes an apparent volume. If 20% is empty, water rises by 80% of its volume. Spacetime behaves similarly, curving in proportion to the mass part.

Earth as an Apparent Volume

Two representations of Earth illustrate how spacetime curvature relates to apparent volume:

• Fig. A: Earth is an apparent volume composed of atoms that are 99.999% vacuum (orbitals, which are empty volumes) and 0.001% matter (nucleus and electrons, which are mass-volumes).
• Fig. B: Shows only the mass-volumes — nuclei and electrons — representing 0.001% of Earth’s total volume. Only the mass-volumes contribute to spacetime curvature. Empty volumes do not affect spacetime.