Saturday, September 2, 2017

Complex Dimensional Geometry

I previously introduced the signed-dimension which generalised fractal geometry to negative dimensions. Is it possible to further generalise to complex dimensional geometry? In other words could we assign a complex value to the dimension of a set in some N dimension space?

What would it look like?
Let's say we have a large extents viewing window and a small cell size, so we have a large and fixed resolution of 'pixels' within the view window. Now we count the occupied cells as we zoom in.
We know from this previous post that for positive dimension sets the count will grow and for negative sets it will shrink. So for an imaginary dimension set we should expect it to neither shrink nor grow, but oscillate. For a complex dimension we expect it to grow according to the real part of the dimension, and oscillate at a rate according to the imaginary part.


 
(a) counting cells for a line as we halve cell size: 1, 2, 4, ..

 
(b) same growth as above. It doesn't have to be a curve to have dimension 1

 
(c) this shape has same growth rate, but with an oscillation: 1, 4, 4, 16, 16, 64, 64,.. 

Above (c) shows a shape that is on average one dimensional, but oscillates relative to the growth. The oscillation repeats each time we divide cell size by 4, this suggests the imaginary component of the dimension should be 2pi/ln(4), which is about 4.53i. As described in the previous post, (c)'s dimension has a positive real part if it is finite size, or a zero real part if it is infinite in extent. In the latter case, it is a purely imaginary dimensional shape. Here are two other examples:

3x3 dust is 2.86i dimensional:

5x5 dust is 1.95i dimensional:

The moving graph shows plots the point count in the view window. Changing the view window size adjusts the shape but not the period of the graph. As you can see the fractals with smaller imaginary dimension are spatially more sparse, which suggests that the imaginary dimension could have a similar role to lacunarity as a description of sparseness, to accompany the fractal dimension. 


To extract the sinusoidal growth pattern from the data you cannot just look at behaviour at a scale extreme, but instead look at the count across all scales. Firstly we need to make the signed dimension more robust to the inclusion of oscillations:

so the signed dimension is:


Secondly we need to extract those oscillations, which give the imaginary dimension:

This represents the rate of oscillation as you zoom in. If the signed-dimension is a and the imaginary dimension is b then the shape is a + bi dimensional. However, since the count can never be negative the shape must also include an a dimensional component, and since the count can never be imaginary, the shape must also include an a-bi dimensional component. This means the shape is multi-fractal. If we were to measure it in standard units then we would say that its size is k1 metres^a + k2 metres^(a+bi) + k2 metres^(a-bi) for some values k1, k2 which are fractal measures.

Acknowledging this necessary multifractal nature, we can give its complex dimension as:
More details are given here.

Sunday, August 20, 2017

Generalised definition for negative dimensional geometry

In the previous post I discussed a few specialist definitions that allow for negative dimensions, and added my own new one. In this post I propose a slightly different definition of fractal dimension that allows for negative dimensions in a generic way, and fits with the use of negative dimensions in physical quantities. I'll call this the signed-dimension to distinguish it, and because it allows any signed number rather than just positives.

The signed-dimension D of a set should be the length exponent at which that set can be measured.

This is different from the standard notion of dimension, which is the length exponent that part of the set can be measured.

The difference is significant. Normally the dimension of a line is said to be 1, but infinitely long lines can't be measured, only counted. The real-dimension is 0, and the same for infinite planes and volumes. It is only finite extent geometry that has the dimension one usually assigns to it. For example:
  • Line segments have D=1 so their lengths can be measured
  • Disks have D=2 so their areas can be measured
  • Points D=0 so they can be counted. 
What about the set of integers on the real number line? The quantity we can measure is the number of points per distance, in this case one per unit length. So the measure is 1 unit^-1, so D=-1.

By this reasoning the set of integer vectors in 2D space can be measured in points per square length unit, D=-2 and similarly for higher order spaces. Notice that these densities can't have dimension less than the negative of the topological dimension, which mirrors the case for positive dimension shapes. Notice also the consistency that in positive dimensions, if you measure assuming D is higher you get 0 and assuming D is lower you get infinity. For negative dimensions the same is happening but the other way around. For instance the number of integer vectors in 2D space per length unit is infinity.

So negative dimensions refer to densities as positive ones refer to quantities. It is somewhat analogous to contravariant (e.g. vectors) and covariant quantities (covectors or one-forms) in physics. But negative dimensions don't have to just be point-like. For instance a 2D grid of lines spaced a metre apart can be measured as two metres of line per square metre, which is 2m^-1. Nor do they have to take on integer dimensions, for example a set of points equidistant along the Koch curve can be measured as the number of points per length^1.26. Another way to think about negative dimensions is that they represent extrapolations of a replacement method, and positive dimensions represent interpolations:



As you can see from the last two, if the replacement scheme acts on lines then the negative dimension version reflects from the 1D position. The replacement scheme can also act on fractal geometry:
   
                                              
                                       
Here the negative dimension version reflects from the 1.26D position of the Koch snowflake. In essence the negative dimension geometry is measuring the density of Koch curves within the infinite extent version of the Koch fractal on the right side.

For replacement schemes, let's call the initial shape in the iterations the tile. Then finite geometry uses the usual Hausdorff dimension:
and the generalised real-dimension adds an extra term:
which is a 0 term on finite geometry since the extrapolate replacement count is 1.

This shows that geometry can be split into a positive dimension (quantity) part and a negative dimension (density) part, and so it is still possible for shapes of infinite extent to have positive dimension, for example an infinite row of disks measures the disk area per metre, so has dimension 1.

In the real world the examples are more stochastic but the definition applies equally. For instance, if you were to measure a country's coastguard capability, you might try to measure the number of coastguards per kilometer of coastline, but this will change depending on what length ruler you use to measure each kilometer. Instead the coastguards can be measured in number per km^1.3 (i.e D=-1.3) where 1.3 is the fractal dimension of a representative piece of the coastline. Just as real coastline is not fractal to infinitely small scales, the coastguards do not cover infinitely large extents. Both are just approximations in the real world.

Aside from the Hausdorff dimension, there are several other dimension measures (box-counting/Minkowski dimension, information dimension, correlation dimension), there are all generalised in the so called q-dimension formula, normally written:
where mu is the natural measure, which is 1/n when that cell of width epsilon is occupied, and n is the number of occupied cells in total.

We can modify this to give a generalised version of the signed-dimension:
where the positive dimension component is:
and the negative dimension component is:
and where:

q=0 is the box counting/Minkowski dimension, q=1 is the information dimension and q=2 is the correlation dimension.

In all cases we can either write the dimension of a set directly, e.g. -0.26D, or leave it in its component form, e.g. 1-1.26D for clarity.

Thursday, August 17, 2017

Dimension-reducing Timelike Curves

We all understand geometry of integer dimension and fractal geometry describes shapes with positive fractional dimension, but there are only a few descriptions on how shapes could have negative dimensions. In fact I know of three.

1. A negative dimensional space as described by V Maslov considers gaps in a solid as negative dimensional, presumably a solid missing a line is -1 dimensional and a solid missing a plane is -2 dimensional. This seems to be incorrect as it is confusing measure with dimension. If anything one might think of the holes as having negative measure (relative to the solid).

2. A less scientific answer is to say something has negative dimension if it appears to fade away as you look more closely. For example getting closer to a cloud, or to a face painted in pointilism, or perhaps on very close inspection a smile disappears and becomes just connected cells. In other words, for something to reduce in intensity as you look with more detail, you need that something to be a concept rather than just a raw set of points.

3. Benoit Mandelbrot had a definition of negative dimensions when you are describing the intersection of different shapes under a small amount of noise. The intersection of a point and a volume is 0 dimensional, a point and a plane is -1 dimensional, a point and a line is -2 dimensional and between a point and a point is -3D. The idea is that as you look more closely the likelihood of intersection within a reducing size ball reduces at different rates for these cases. It is a nice idea which removes a special case in a certain set operator equation, but it assumes noise so is a somewhat specific definition. It also creates some odd cases such as a 1D plane (if you a intersecting it with a plane) and intersecting more than two sets gives unbounded negative dimension. In fact it is a specific type of idea 2, as the dimension is being defined for a concept (the intersection of sets) rather than just sets.

An alternative means of creating a dimension reduction, which can extend as far as negative dimensions, is to use time-like curves, i.e. using the fact that square distance of time quantities is negative. Here we assume the set will embed in the standard n+1 Minkowski space-time. The proper length of time-like curves can decrease with increasing detail level if they tend toward light-speed motions:

Reducing dimension time-like curves
This particular family of curves is the 1+1D Minkowski equivalent of a Koch curve in 2D Euclidean space. In fact, we can think of the Koch curve, the Weierstrass function and this curve to be the same type, only differing by whether the second dimension squared is positive, zero or negative respectively. And accordingly the larger angles increase, make no difference and decrease the dimension respectively, from a 1D line.

The same curves and same dimensions also exist on the complex plane, provided that it is with respect to the real-length of the curve, defined as:



In 2+1 space-time the results represent a point moving around in 2D and are perhaps more visually interesting. Here I follow the replace pattern of the Koch curve in 2D space, but the bend angle doesn't increase the curve length by a fixed percentage (like the Koch curve), instead it is achieved by boosting the motion in order to decrease the proper time by a fixed percentage. As a result the dimension decreases and the measure is of the proper time, meaning we quantify the period of the motion as the time according to the point, in seconds^D where D is the dimension.
This effect is a bit like the internal scattering of light in some medium. Lower dimensions are like there are a greater range of particle sizes, with few that cause large reflections and many that cause tiny course changes. D=1 is like all particle sizes (or scattering angles) are the same.

So far we have said that the proper time of the trajectory has some dimension b. We could also piggyback the a+bD syntax for Lorentzian space-time to say that the overall geometry has dimension 0+bD, i.e. the temporal dimension is b and the spatial dimension is 0. If we moved a line along this trajectory then it would be 1+bD geometry.  

Unlike Euclidean space fractals, the paths of these fractals needn't appear rough, here is an example that is C(1) continuous, built out of circular arcs:
It has some similarity to the Hevea project curves, but the above paths are actually moving points with a fractional dimension, despite their smoothness. 

In the next post I give a general proposal for negative dimension geometry for the more common Euclidean space. 

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For reference here is the 1+1D version of the 2D Levy C curve, which is the Blancmange curve in Newtonian space-time:
and the 2+1D version:
Notice that the 0D curve appears like two circular arcs. I don't know if they are perfectly circular or indeed perfectly smooth, but it is certainly close in both cases. There is certainly something interesting about the fact that the smooth path has dimension 0, it is perhaps even obvious that it should, since light-like motion has zero proper time so is 0D. 


Saturday, August 12, 2017

Mach's principle

Mach's principle is fascinating. If General Relativity does obey it then GR is telling us something much more profound than just adding gravity and bendy space-time to special relativity.

Mach's principle is that our idea of what a rest coordinate frame is comes from the motion or shape of the bodies in the universe. So if all of the universe was rotating, our rest coordinate frame would also be rotating and it would be equivalent to the whole universe being at rest, as we see it now. Equivalently, if you spin around in space, your arms only separate with centrifugal force if there are stars that you are spinning relative to.

This is a wonderful idea that is telling us something more than Special Relativity. It is saying that space-time itself is only a product of mass-energy. It is not just that mass-energy warps space-time, space-time exists because of a distribution of mass. So flat (Minkowski) space-time does not result from the absence of mass, it is the approximation of space-time when mass is fixed and very far away. To put it a third way, matter doesn't sit on an infinite fabric, the fabric exists only in relating the motion of matter.

The only problem is that, even 100 years after GR, people can't seem to agree whether GR really is 'Machian'. However, on inspection it seems to me that it is:

Wheeler-Einstein-Mach spacetimes

An important paper is Wheeler-Einstein-Mach spacetimes:

"We define the Wheeler-Einstein-Mach (WEM) spacetimes to be those which contain a closed Cauchy surface, are inextendible, and satisfy field equations with a well-posed Cauchy problem, We show that a WEM spacetime can be reconstructed from the "York data" on any given closed (constant mean curvature) hypersurface contained in that spacetime. This result is the strongest and most precise statement to date of Wheeler's version of Mach's principle."

While the Machian view is that the inertial frame can be reconstructed by the distribution of mass-energy in the universe, the authors don't argue over exactly what this means:

"To decide, within these prescribed limits, exactly what should be labeled "mass energy" and used as the Machian input, we eschew philosophical considerations and choose what gives us a nice theorem, i.e., we choose the York data"

they just use the York data:
"Define the "York data" Q, of a given spacelike hypersurface S is composed of the conformal intrinsic geometry, the conformal (transverse traceless) extrinsic geometry, the conformal nongravitational and the mean extrinsic curvature scalar"  (paraphrased)

So based on this, they have found that this large class of GR space-times show no rotation of their universe from their preferred (rest) coordinates. Which is consistent with the idea that your coordinates rotate with the universe:

"Absolute rotation: None of the WEM spacetimes has an absolute net rotation in any well-defined sense."

People have cited the Godel universe as a counter-example, but this paper refutes it clearly:

"Local vorticity (of the sort found in the non-WEM Godel universe and also in the WEM Ryan spacetimes) does not constitute net rotation of the universe in any sense."

The idea that the Godel universe is local vortices is backed up by the wikipedia article on the Van Stockum dust solution:
"Note that unlike the Gödel dust solution, in the van Stockum dust the dust particles are rotating about a geometrically distinguished axis"

Frame Dragging

In the Machian view, the coordinates would rotate with a galaxy if there were no other in the universe. However, since there are others far away, it makes sense that coordinates would rotate partially with the galaxy, and to a lesser degree as one moves out towards the rest of the universe. This does happen and is called frame dragging or the Lens-Thirring effect. Again supporting the Machian idea. 

The linear acceleration equivalent is described in this very clearly written paper, again supporting the Machian principle.

Rotating Mass Shell

Also there is this paper: Induction of correct centrifugal force in a rotating mass shell
It shows that for a correctly shaped shell over fixed Minkowski space, the flat space-time inside the shell will rotate with the shell:

"Mach’s idea of relativity of rotation is confirmed for a shell-type model of the universe by showing that flat geometry in rotating coordinates, realising correct Coriolis and centrifugal forces, can be continuously connected through a rotating mass shell with not exactly spherical shape and latitude-dependent mass density to an asymptotically Minkowskian outside metric. The corresponding solutions of Einstein’s field equations are given to second order in the angular velocity w but it is plausible that the problem has a solution to any order of w"

I find this a very clear vindication of the principle. Even though it requires a specific shell shape, I expect that requirement becomes less sensitive if you replace it with a more disperse and distant universe.

Ozsvath-Schucking space-times

This metric is also cited as a refutation of Mach's principle. The space-time seems to be parallel gravitational waves. It is said to be an anti-mach metric because it contains no mass but is not just flat Minkowski space. This seems like a weird reason to claim to be non-Machian, for three reasons:
  1. Mach's principle doesn't claim that a universe lacking matter should be Minkowski, since that would preference inertial (rather than nonlinear) coordinates. It claims that a universe lacking matter has completely unconstrained rest coordinates.
  2. As discussed in the WEM spacetime paper above, the problem with infinite space-times is that they exclude a light-cone at infinity, and this may well assert a constraint on the rest frame. Hence the WEM spacetimes are compact.
  3. Mach's principle that the rest frame is derivable from the mass distribution alone was before anyone knew about gravity waves, it seems OK to me to include the dynamics of these waves in resolving the universe rest frame. Besides, gravity waves are a form of non-localised energy. The principle can remain, that if there is no mass or energy (including gravitational waves) then there can be no preferred rest frame (linear or otherwise).

Conclusion

So the way I think of it is that space-time stretches between the masses and so it is impossible to notice any overall motion of the full set of masses. For example if all of the stars in the universe were actually oscillating laterally all at the same time with an amplitude of one metre, then space-time would also oscillate with it, and so the motion would be unobservable, and this goes for any motion. Note that you don't have to invoke magic for this, you need to attach rockets to all the stars and they will give positive and negative pressure values in the stress-energy tensor which bend space-time correctly to physically produce the oscillation.


So while Newton's relativity and Special Relativity both have no privileged inertial frame (linear velocity), in GR there is no privileged coordinate frame at all, linear or otherwise. Nature is not preferring linear motions, only geodesic ones. And ultimately our idea of what it means to be at rest comes down to a sort of average of the stars around us. Everything is indeed relative.

The main problem that seems to prevent people from accepting the Machian principle is that Minkowski spacetime prefers zero angular velocity and acceleration. Einstein understood this to be a problem with the metric at infinity, and that it is required to be constant. The issue with Minkowski spacetime is also expressed in this paper, quoting:

"The necessity of introducing an extended model of the Minkowski spacetime, in which a globally empty space is supplied with a cosmic mass shell with radius equal to its own Schwarzschild radius, in order to extend the principle of relativity to accelerated and rotational motion, is made clear."

"In the general theory of relativity the significance of the Minkowski spacetime is that it is used as the asymptotic metric outside a localized mass distribution, for example in the Kerr spacetime. This means that absolute rotation is introduced into the general theory of relativity through this choice of boundary condition when solving Einstein’s field equations."

and here supports the idea that there will be perfect inertial dragging for our universe:

"A meaningful boundary condition for flat spacetime is to introduce a massive shell that represents the cosmic mass inside the shell. As shown in the previous section the mass inside the lookback distance of our universe has a Schwarzschild radius equal to the lookback distance. Hence, it is natural to impose the boundary condition that the asymptotically empty spacetime is replaced by the boundary condition that there is a mass shell at the lookback distance with radius equal to its own Schwarzschild radius and mass equal to the cosmic mass inside the lookback distance."

Inertia

Another point of contention is how Mach's principle effects the inertia or mass of objects. This is worth investigating, by looking at how linear and angular motion changes as the mass of the remaining universe drops down towards zero. In each case we can either have the background universe exclude the observer's mass (a) or include it (b).

Linear oscillation

In this case we want to oscillate by applying a sinusoidal force of fixed amplitude. The observer is pictured as the large sphere and the remaining (background) universe is simplified to just a single small sphere.
(a) In this case, as the observer applies its sinusoidal force (e.g. by emitting radiation, of zero mass), its large mass compared to the rest of the universe causes substantial frame dragging and the universe mainly follows the same sine wave. Consequently, the observed motion relative to the background universe has reduced.  

The rest point in this set-up is I think roughly at the centre of mass of the two bodies, which is much closer two the observer body. As such, the motion of the two bodies relative to the centre of mass looks like (with amplitudes scaled up):
(b) Relative to this fixed point, the motion is also smaller than for an infinite mass universe. 

In both cases the motion is less than expected and the observer may think that his mass has increased. For zero background mass, the lateral forces cause no motion.

Angular motion

(a) For a low mass background universe any angular torque will cause the background universe to rotate somewhat with the observer, so a larger torque is needed to attain a target angular velocity. However, the frame at the observer will rotate mostly with the observer, so the proportion of this relative angular velocity that causes centrifugal acceleration on the observer reduces.


(b) We define the rest frame as being the point where frame dragging of observer and universe cancel out (see pale blue lines). In coordinates where this cancel point is at rest, the torque required to achieve an angular velocity relative to this rest frame increases with decreasing universe mass. In this case, the rest point gets closer to the observer as the background universe mass decreases. So once you achieve the required angular velocity relative to the whole universe, your centrifugal acceleration would probably be similar as the background mass decreased. 

Conclusion

While it may appear to the observer that its mass and/or inertial moment increases with a diminishing background universe, this is not really the case. What is happening is that the effective mass/inertia are changing because these properties are relative to the universe's mass/inertia. As long as effective mass/inertia and absolute mass/inertia are conceptually distinguished the discussion of inertia remains clear.
In both choices of rest coordinate (a) and (b) the effective mass increases because forces applied to the observer act increasingly and oppositely on the universe rather than the observer as the universe mass diminishes.

Both examples demonstrate Mach's idea, that inertia only exists in the presence of a background universe.

The centrifugal force question is a bit more subtle. You certainly would feel less centrifugal force as the star masses decreased if you measured your rotational velocity relative to the stars, but if you measured it relative to the rotation of the universe including yourself, then centrifugal acceleration may well stay constant. What reduces is your ability to achieve that angular velocity, in other words your effective moment of inertia increases. 

Scale Dragging and Dark Energy

Is it possible that the expansion producing negative energy termed dark energy is in fact the apparent opposite action on the universe due to contractive frame dragging?

Imagine that the universe is a scattering of stars with no initial velocity, then they will contract under gravity. But if the entire universe is contracting then I would expect frame dragging to be contracting the reference frame as well. Therefore, relative to this reference frame the universe should have no net contraction. In fact, if the galaxies contract less than clusters due to their spinning, then the rest frame is contracting faster than the actual contraction of the galaxies, and in the rest frame we will see the universe as expanding. The process known as dark energy. 

Thursday, July 13, 2017

general relativity

When you toss a coin, the coin falls down because there is less time at lower altitudes, and time passes slightly more slowly. If we plot altitude against time on a sheet then we expand time horizontally at lower heights, the sheet will splay outwards as shown below. A coin tossed upwards simply follows a locally straight path on this sheet, like an ant walking along it in a straight line. 
We stretched rather than contracted the time direction because we are representing the time dimension spatially, and they have opposite metric signatures. Space and time don't actually bend into a third dimension like the sheet, the curvature is just intrinsic to the space and time; you can just think of time as being less dense at lower altitudes.

The field equation of relativity holds at all points in space and time and is
where Gij is the Einstein tensor describing the bending of space-time, k is a constant and Tij is the stress-energy tensor which represents the density and pressure of matter at that point.

Gij and Tij are both 4x4 tensors (very similar to matrices) but they are symmetric and this means they represent a scaling along four orthogonal principle axes. For the spatial dimension this can be seen as stretching a sphere into an ellipsoid:


If we work in this principle inertial frame (the rest frame) then the field equation just has diagonal entries, e.g:
ρ is the density of matter and px,y,z are the pressures, which are equal in most idealised cases, such as an idealised fluid.
Scaling by k gives the Einstein tensor Gij, where each diagonal entry represents the scalar curvature of the 3D volume orthogonal to that entry's axis. For example the first entry means the 3D spatial curvature is kρ since the spatial volume is orthogonal to the time axis.

This 3D scalar curvature (or 3D Ricci scalar) is proportional to the difference between the surface area of a small sphere around that point, compared to the expected surface area of a sphere in Euclidean space. A positive scalar curvature (as is the case when mass is positive) has a smaller than expected sphere area, and parallel paths converge. In the case of the first entry this means that mass causes space to have spherical curvature, like a 3-sphere.

This is the simplest explanation of General Relativity; the stress-energy tensor represents the scalar curvature of the 3D volume orthogonal to each principle axis.

For earth, the density ρ is 5,500 kg/m^3 and since k is the rather small 2e-43, the spatial curvature G00 is 1.1e-39. This results in the earth being 2mm smaller in radius and with 35 fewer hectares of ground than if it were in flat space. 

A closer look

However, this definition isn't very helpful in telling us how space and time distort with each other, since the scalar curvatures on each axis combine together in their effects. In order to get actual differential equations we expand out the field equation definition to:

where Rij is the Ricci tensor which represents the 4D curvature of space-time and R is the 4D Ricci scalar. Working in the mass's rest frame, the Ricci tensor is just a linear transform of the stress-energy tensor:

This transform can be graphed in 2D when the pressure is equal in all directions (using units where k=1):

For any given mass and pressure of a body the blue arrows show the directions of increasing contraction with respect to time and with respect to space. Lets look at this curvature more carefully.

Each element in Rii defines the rate of change of contraction of a small orthogonal 3D volume V with respect to the element's axis. For R00 this 3D volume is just the spatial volume (written as Vxyz): 
as given by John Baez's relativity tutorial. For R11 it is a small time period t multiplied by the perpendicular area yz:
Note that Vtyz isn't exactly a volume, while increases in the yz directions increase it, increases in the time period do the opposite, due to the negative metric signature:
and equivalently on the other two spatial axes. This is for a volume at rest, otherwise we subtract volume acceleration in flat space from the left hand side. The negative sign is because these are contractions.

In standard units the equations just contain a few more constants:
and
Noting that the first term in the stress-energy tensor is actually energy density rather than just density, and since e=mc^2, that is why we use ρc^2.
So for these "normal" units of density and pressure, the pressure has almost zero effect and also changes with respect to space are tiny compared to those with respect to time. 
For example, for earth ρ is 5,500 kg/m^3 and earth's pressure is 2e11 Pascals, so the pressure term is 4e-8% of the energy density term. 
So the rate of volume contraction is 0.0005% per second squared. 
whereas the rate of Vtyz contraction with distance is 5e-21% per metre squared.

Therefore curvature per metre squared is roughly 1e-15% of the curvature per second squared, for earth.

Example of space above earth:

The Ricci tensor in space is zero. This means no volume Vxyz  acceleration with time, and no time x area (Vtyz) acceleration with distance x. This does not make space Lorentz flat, it can still bend and preserve 3-volume and that is what it does near earth:
- With respect to height, the horizontal area contracts at lower altitudes (being 35 hectares less than in flat space at ground level), and the other coordinate must expand to compensate (we draw it spatially as expanding), but since this is the negative signature time period, it must also reduce, and it does, causing gravitational time dilation.
- With respect to time an initially static volume's height expands since gravity is slightly stronger closer to earth, but its width and length get shorter, as it moves downward where area is slightly contracted.
In both cases the expansion and contractions cancel out to give zero volume change.

This diagram shows how objects move above ground, causing vertical expansion, and below ground, causing vertical contraction. In solids the pressure forces resist the inwards pull. Therefore relative to these straight yellow lines, the ground surface is in fact accelerating upwards.

Spatial curvature is much less pronounced than curvature with time. This bend is irrespective of speed.
  


For more details try out my cheat sheet:

and some other helpful links: