General relativity and the block universe and pseudo-Riemanian spacetime

 In relativity, the answer is not exactly. What is negative is the squared spacetime interval under one common sign convention.

In ordinary Euclidean geometry, the distance between two nearby points is

[ ds^2 = dx^2 + dy^2 + dz^2 ]

which is always non-negative.

In flat Minkowski spacetime, one common convention is

[ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2. ]

Notice the minus sign. Because of it, (ds^2) can be:

• Negative (timelike separation)
  
• Zero (lightlike separation)
  
• Positive (spacelike separation)
  

For two events occurring at the same spatial location but at different times,

[ dx = dy = dz = 0 ]

so

[ ds^2 = -c^2dt^2. ]

This is negative. That's what you were noticing in our earlier discussion.

However, physicists often define the proper time (d\tau) along a timelike worldline as

[ d\tau^2 = -\frac{ds^2}{c^2} ]

(using the sign convention above).

Since (ds^2<0) for timelike motion,

[ d\tau^2 >0, ]

and the elapsed proper time (\tau) is a perfectly ordinary positive quantity.

What about General Relativity?

General relativity replaces the flat Minkowski metric with a curved metric:

[ ds^2 = g_{\mu\nu}dx^\mu dx^\nu. ]

Locally, at every point, spacetime still has the same Lorentzian signature as Minkowski space. The usual signatures are either

[ (-,+,+,+) ]

or

[ (+,-,-,-). ]

The choice is merely convention.

• With ((- + + +)), timelike intervals have (ds^2<0).
  
• With ((+ - - -)), timelike intervals have (ds^2>0).
  

Both describe exactly the same physics.

From a geometric perspective

Riemannian geometry uses a metric that is positive definite. Distances are always positive.

Spacetime in general relativity is actually a pseudo-Riemannian (or Lorentzian) manifold, not a purely Riemannian one. The metric is not positive definite, which is why intervals can be positive, negative, or zero.

So in the block universe picture of general relativity:

• The spacetime interval (ds^2) between timelike-separated events is often negative (depending on convention).
  
• Proper time along an observer's worldline is positive.
  
• The "negative interval" is not a negative amount of time. It is a consequence of the Lorentzian geometry that distinguishes the time dimension from the spatial dimensions.
  

An elegant way to think about it is that Euclidean geometry treats all directions alike, while Lorentzian geometry gives the time direction a different sign. That sign difference is precisely what creates light cones, causality, and the distinction between past, future, and elsewhere.

The number line and a circle as one dimensional geometry

 This is a subtle question because the answer gets at the difference between dimension and topology.

Both a straight number line and a circle are one-dimensional spaces. At any point on either one, you can move in only two directions: "forward" or "backward" along the space. So dimension alone cannot distinguish them.

The difference is in their global structure.

The Number Line

The ordinary number line (\mathbb{R}):

• extends infinitely in both directions,
  
• has no loops,
  
• has no beginning or end, but it is unbounded,
  
• can be cut at any point and still remain connected on each side.
  

Visually:

<------------------------>

Every point corresponds to exactly one real number.

The Circle

A circle (S^1):

• is finite in extent,
  
• loops back onto itself,
  
• has no endpoints,
  
• is bounded but has no boundary.
  

Visually:

   _____

 /       \

|         |

 \_______/

If you travel far enough in one direction, you eventually return to where you started.

Why a Circle Isn't Just a Curved Line

You can bend a line into all sorts of shapes without changing its fundamental topology. For example,

~~~~~~~~~~

or

/\/\/\/\/\

are still topologically equivalent to a line.

But when you connect the ends together, something fundamentally changes. You create a closed loop.

One way mathematicians detect this is by asking:

Can the space be continuously deformed into another without cutting or gluing?

A line cannot be deformed into a circle because forming the circle requires gluing two distinct points together.

Coordinates Reveal the Difference

On the number line, positions are labeled by real numbers:

[ x \in (-\infty,\infty) ]

On a circle, positions are often labeled by an angle:

[ \theta \in [0,2\pi) ]

but with the identification

[ 0 \equiv 2\pi. ]

This identification is the key. Two different coordinate values represent the same point.

In a sense, a circle can be viewed as

[ \mathbb{R}/\mathbb{Z}, ]

which means taking the real line and declaring every integer shift to be equivalent:

[ x \sim x+n. ]

The line becomes periodic.

Connecting This to Your Earlier Question

Notice how dimension alone is not enough to characterize a space:

• A point is 0-dimensional.
  
• A line is 1-dimensional.
  
• A circle is also 1-dimensional.
  

The line and circle have the same local dimensionality, but different global topology.

Similarly, in relativity:

• Minkowski spacetime and a curved spacetime can both be 4-dimensional.
  
• What differs is the geometric and topological structure, not the dimension count.
  

So the shortest answer is:

A circle and a number line are both one-dimensional because locally they look like a line. The difference is that the number line is open and infinite, while the circle is closed and periodic—the ends are identified with each other, giving the space a different topology.

Zero dimensional space

 Yes. In both mathematics and geometry, a single point can be considered a space, and it is indeed zero-dimensional.

Here's the intuition:

• A 1-dimensional space needs one coordinate to specify a location (for example, (x) on a line).
  
• A 2-dimensional space needs two coordinates ((x,y)).
  
• A 3-dimensional space needs three ((x,y,z)).
  

A single point requires no coordinates to move within it, because there is nowhere to move. It has:

• no length,
  
• no width,
  
• no height,
  
• and no internal direction.
  

So its dimension is 0.

There is a nice progression:

Dimension	Example	Degrees of Freedom
0	Point	0
1	Line	1
2	Plane	2
3	Ordinary space	3

A useful way to think about dimension is as the number of independent directions available for motion. At a point, there are none.

This connects intriguingly with the discussion we've been having about spacetime. In Minkowski spacetime:

• An event is represented by a single point in four-dimensional spacetime ((t,x,y,z)).
  
• The event itself is 0-dimensional.
  
• The collection of all events forms the 4-dimensional spacetime manifold.
  

So when physicists say that an event (such as a flash of light occurring at a particular place and time) is a point in spacetime, they are treating it as a zero-dimensional object embedded in a four-dimensional space.

There is also a deeper geometric perspective. Just as a line can be thought of as a collection of points, some mathematicians view dimensions recursively:

• a point is a 0-dimensional space,
  
• a line is generated by moving a point,
  
• a plane by moving a line in a new orthogonal direction,
  
• ordinary space by moving a plane in a new orthogonal direction,
  
• spacetime by adding a time-like dimension to space.
  

This is very similar to the analogy you were making earlier about going from a line to a plane by introducing an orthogonal axis. The "dimension below a line" is precisely a point. The question of what would be "below a point" leads into more abstract mathematics, where some people speak informally of the empty set as having dimension (-1). That's not a physical space, but it plays an important role in topology and geometry.

So, in short: yes—a single point is a perfectly valid space, and it is the canonical example of a zero-dimensional space.