# In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?

I've been working on studying Special Relativity and General Relativity for the past few years. As I think we all know, GR gets a lot more complicated than SR, and my knowledge is limited. I am very familiar now with Minkowski space/time diagrams. I have been working with equations for a (Spatially) One-Dimensional Universe, as is common when working with Minkowski Space.

For clarity, this (1D) universe has one spatial dimension and one time dimension, with each observer defining different "x" and "y" axis-angles depending on their relative speeds. (In other words, each relative observer has their own line representing time and distance from their perspectives if you disagree with my first description)

1D Flat Space-time without any gravitational effects would look like this diagram I made in Desmos, where the Green represents observer "Green" and the Red represents observer "Red". The Blue dashed lines represent light waves. (This Diagram is from my pet project - a fully functioning, mathematically correct description of Minkowsi Space-Time that I've made in Desmos with variables that can be altered representing relative velocity, acceleration, and other concepts)

Now, we know that humans cannot see or imagine (very well) anything in dimensions higher than 3. So, using just 1D space, is it possible to accurately and mathematically describe and graph Minkowski space with the addition of Gravity? I've only seen crude "conceptual representations" of this that always contain a disclaimer that it is an oversimplification. Note: I know that gravity does not "work" the same way in less than 3-Dimensions when it comes to orbits etc., that is fine - we still theoretically should be able to create "gravity-like" effects in a 1D spatial universe in Desmos or another graphing program by applying the same concepts and equations from GR to lower dimensions

If this can't be done - Why not? Can't we either warp the 2D graph that I've provided to make sensible claims about Gravity in a "1D" universe, or expand it into a 3D graph if the warping of space-time requires an extra dimension? I'm told it does not require an extra dimension but I am a not quite sure:

## My oversimplified conceptual attempts look like this:

Option#1.) Here is a spatially 1D, 2D Minkowski space-time diagram, warped into the 3rd dimension to represent gravity. First, a crude diagram of what I mean by warped into the 3rd Dimension, and then a "top-down" view of the same diagram as before with the warping added. The black represents a body of mass. Note: Option#1 seems to imply speed of light is locally the same but a distant observer will see light as moving slower closer to mass - which I believe to be correct

Option#2.) Here is a spatially 1D, but strictly 2D Minkowski space-time diagram, where gravity warps the "present" line of space. Note: Option#2 appears to be 3D, but it is not. The lines actually just are bent on a 2D plane, giving the illusion of 3-D.

Is (Option#1) OR (Option#2) correct and can be made into a fully-functioning concept that accurately tracks time, space, and light's movements in a gravitational field(s)?

OR - (Option#3) Is there a third correct option that I'm missing?

OR - (Option#4) Is it for some reason impossible to create a mathematically accurate, understandable, lower dimensional Minkowski model of space-time including a lower-dimensional representation of gravity?

Please do not include "conceptual sketches" that are not accurate representations of GR as examples - That is not what I am after, I am looking for a true correct representation of GR for a spatially 1D universe.

Please do not include an overabundance of mathematical equations or technical jargon in your answer without a straightforward answer to Options#1, #2, #3, or #4 as this is a visually based question, and I am looking for a straightforward answer of whether or not this can be done accurately and how it should look

Please do not answer if you are uncertain of your answer as I do not want to become any more confused on the topic and want to pursue modeling this correctly.

## Edit in Response to Answer from Benrg

(Option#5) If time slows down near mass, and the time dimension is uniformly stretched vertically, while the horizontal space dimension remains normal (unstretched), can we construct an accurate non-grid model as I've attempted again using dot density as suggested, below?

If the distance lines are, for example, light-seconds - then the Squares can be representations of seconds from the perspective of an observer in various parts of the gravitational field. (The mass runs vertically along the center of the graph, passing through 0,0)

Same as above but including light waves passing towards and away from mass, through gravitational field:

The horizontal lines are an arbitrary artifact - they convey the "present" lines of a stationary observer relative to the mass but they no longer are a fixed amount of time apart, because the length of time varies depending on where in the gravitational field we are.

Is this option correct, using a "gravity-like" concept for 1+1D according to your interpretation?

• possibly useful: physics.stackexchange.com/q/1417 Commented Aug 12 at 23:08
• possibly useful: ("Sector Models" by Corvin Zahn and Ute Kraus) spacetimetravel.org/sectormodels1 Commented Aug 12 at 23:20
• No, you can not draw a global Minkowski diagram for the same reason that you can't draw a global flat map of Earth, either. The correct solution is given by differential geometry where each point on the manifold has an infinite number of local Minkowski diagrams attached to it. Commented Aug 12 at 23:21
• Thanks FlatterMann. You can, however, draw a 3D globe of the earth, so this isn't a great example. Can't we make a 3D model of a 1D universe in the same way? That is the question after all. Commented Aug 13 at 1:40
• And... if each point on a line (1D space) was associated with a 2D plane (Minkowski diagram), as you say, that merely forms a 3D construct - so I fail to see how this can't be modeled? Commented Aug 13 at 1:48

It's not clear that GR generalizes to 1+1D. In two dimensions the stress-energy tensor has more components than the curvature ($$3>1$$), so it's not obvious what the GR field equation should look like. That isn't the end of the world, because a lot of what people call "general relativity" is just calculating geodesics on a fixed spacetime manifold, and you could do that in 1+1D even if the shape of the spacetime isn't justified by a 1+1D theory of gravity. That aside:

In option 1 and option 2, it looks like you've just applied a distortion transformation to the $$x$$ or $$y$$ coordinate of your flat-spacetime diagram. If so, you're just plotting the same old flat spacetime in a funny way. Coordinate transformations don't change the intrinsic geometry.

Your other illustration, , doesn't seem to match what's meant by curvature in general relativity. If you draw a triangle on a piece of paper, its angles will add to 180°. If you bend the paper, the angles will still add to 180°. The paper is not truly curved unless the angles of at least some triangles on it add to something other than 180°. You can make a sheet of paper truly curved by dampening and then drying it. A truly curved sheet can't be flattened without tearing it.

A correct diagram of curved 1+1D spacetime probably can't look much like your diagram of flat 1+1D spacetime. In your flat-spacetime diagram, the solid lines are the 1+1D version of a Cartesian coordinate grid: evenly spaced parallel lines that intersect each other at right angles. In a general curved space, it isn't possible to draw such lines. Parallel lines may not exist (depending on how you define them), and in any case will not stay a constant distance apart when extended, and that's before requiring that they meet at right angles.

If you nevertheless draw a grid of some sort on the manifold, you then have the problem of how to plot it. If you use the grid coordinates for the plot, then the grid will look like a regular rectangular grid and won't visually indicate anything about the curvature.

One solution would be to use Tissot indicatrix circles. Any 1+1D metric can be written at least locally in the form $$ds^2 = f(t,x)^2 (dt^2-dx^2)$$, which looks like Minkowski space except for the $$f(t,x)$$ scale factor. You could use the $$(x,t)$$ coordinates for plotting, drop the coordinate grid, keep the dotted diagonals representing lightlike geodesics, and add some circles whose radius is inversely proportional to $$f$$ (so that they all have the same physical area).

That's a mathematically accurate representation of the curved manifold. Whether it's understandable is up for debate. The Mercator projection with indicatrices looks like this:

By Tobias Jung, CC BY-SA 4.0, from the site linked earlier

That's a sphere, or a slightly oblate spheroid, but it's pretty hard to tell from the circles. Geodesics of the sphere are great circles, and look sinusoidal on the projection, but that's hard to tell too.

• Thanks for the well thought explanation. I believe it might be possible to model this, in a way, based on your answer. If time slows down near mass, the time dimension can be uniformly stretched vertically, while the horizontal space dimension remains unstretched. This is why you say the grid will still appear as a grid. However if we use dots to represent the a fixed amount of time instead of a grid, that grow further apart along the vertical lines as they get closer to the mass, then that might be an accurate representation, unless I'm still missing something. I will provide another diagram. Commented Aug 13 at 2:49
• @Mr.Green It would be possible to show $f(t,x)$ by the density of dots instead of the radius of circles. Arbitrary 1+1D geometries can't be written in the form you're suggesting ($ds^2=g(x)^2 dt^2 - dx^2$), but a radial+time slice of the 4D Schwarzschild geometry can be. Commented Aug 13 at 3:28