Can you imagine general relativity without a continuous spacetime manifold? Both concepts seem to be inseparable, but I will show here that Lorentzian manifolds are even not compatible with GR. The principles of GR are describing worldlines, but not the vacuum points between worldlines.

After the development of Einstein's special relativity, Minkowski presented in 1908 his model of a continuous spacetime manifold. Later it was Einstein himself who introduced a curved spacetime manifold for the mathematical description of gravity. This mathematical model of a curved spacetime manifold was considered an inseparable part of general relativity, but to date it has not been possible to harmonize it with quantum mechanics.

In the following it will be shown that the concept of a Lorentzian manifold does not only not comply with quantum mechanics, but that it is even not compatible with the principles of GR itself, and that spacetime is limited to worldlines, at the exclusion of vacuum points between worldlines, because

I. Vacuum is not defined,

II. There is no coherent interval concept for Lorentzian spacetime,

III. Coordinates of observers are continuous manifolds, but they have no Lorentzian metric.

I. Vacuum is not defined by GR

The issue is shown very clearly by special relativity, with its two postulates:

- The laws of physics are the same in all inertial reference frames.

- Speed of light is measured with the same value c in all inertial reference frames.

Manifestly, both postulates are talking about inertial reference frames (and also about lightlike phenomena), but not about the vacuum between worldlines. In particular, time evolution is assigned to worldlines, but it is not defined for vacuum which seems to be a timeless void. Vacuum is defined by quantum physics (and possibly by cosmology with respect to dark energy), but neither by SR nor by the gravitational spacetime curvature models of GR.

II. GR has no coherent spacetime interval concept - spacelike intervals would imply square roots of negative numbers

Example: A spaceship is traveling from A to B in 5 years, the space distance between A and B is 3 light years - what is the spacetime interval between A and B?

Physics provides 4 contradicting answers: the spacetime interval could be 4, 4i, 16 or -16, depending on our choice of signature (+,-,-,-) or (-,+,+,+), and of extraction of the root or not. But whatever is our choice, we get imaginary spacetime distances (or negative squares) in one sense, either for timelike or for spacelike intervals.

This dilemma is due to an intrinsic deficiency of Lorentzian "manifolds" whose definition is limited to worldlines, at the exclusion of vacuum points between worldlines, and this is why spacelike spacetime intervals are imaginary and meaningless.

III. Coordinates of observers are continuous manifolds, but they have no Lorentzian metric.

What about the coordinates of observers? They seem to be continuous manifolds, and vacuum points between worldlines seem to be defined. The answer is yes, they are manifolds, but they are no pseudo-Riemannian manifolds. As soon as we try to introduce a Lorentzian metric onto the coordinates, the continuity get's lost, only worldlines are remaining, at the exclusion of vacuum points between worldlines, and only worldlines are transforming from one observer to another.