Martian Off-equator Space Elevator

A decades old science fiction theme (or dream) has been the space elevator. This is a (really strong) cable running from the equator through geosynchronous orbit and then on to a suitable stabilization mass. Vehicles running up and down the cable would provide an efficient, low cost, route to planetary orbit and beyond. The technical challenges are immense requiring materials far beyond current state of the art but perhaps possible.

Building such an elevator on Mars presents two additional problems, Phobos and Deimos, the Martian moons which are directly in the path of an equatorial space elevator, unlike Earth’s moon. The blocks below indicate the extremes of the moons’ orbits. One suggested scheme is to create an elevator that sways back and forth under power to miss each moon as it transits, every 11 hours for Phobos and every 131 hours for Deimos. This is recipe for disaster, constantly dodging near misses and hoping nothing goes wrong or wears out in the maneuvering system. Initially it seems reasonable to locate the ground end off the equator far enough that the cable is above the maximum latitudes of the moons with a large enough counterweight to stabilize the revolution. Unfortunately, while stable, there is no way to actually build it. The traditional equatorial based design is built out from geostationary orbit keeping the up and down masses equal until reaching the ground. This is impossible here. With the elevator plane of revolution over the 45th parallel for instance as there is no stable orbit to build from.

However, it may be possible to build it out from one of the poles.

The main idea is like swinging a weight on a rope around your head and then letting out more rope. As long as you spin the rope fast enough the weight does not fall to the ground. While this is a serious mega-engineering project, just making the cable is probably the hardest part. Start with a large, massive structure at the rotational pole. Think the great pyramid at Giza but much bigger. Vertically through the middle is a hollow shaft topped by an eccentric crank with a bearing that connects to the ground end of the cable. The hollow shaft is driven to spin the cable. A streamlined mass is attached to the end of the cable. At initial launch, the eccentric and short cable assembly are spinning fast enough to keep the mass off the ground. The space elevator is is built out from the center through the hollow of the shaft which would probably of kilometer order dimensions. As mass is added to lengthen the cable and increase the counterweight, angular momentum would slow the rotation but increased centrifugal force due to increasing radius would keep tension on the cable. At first the angular drive from the shaft would need to contend with the atmosphere, hence the streamlining, where the drive has greater leverage. This problem will be reduced once the bulk of the mass is outside the atmosphere. With care it should be possible to balance the slow down with the radius increase to bring the end of the cable to a stationary condition at design altitude. At this time the cable will be moving synchronously with the planet which will eliminate drag and the eccentric will be aligned with the cable and no longer turning. By moving the eccentric ahead or behind by turning the shaft slightly, small adjustments to the elevator’s revolution speed or longitude can be effected as the planet’s rotation pulls or eases on the cable.

The figure illustrates the geometry for a 40,000km cable. This assumes negligible cable mass compared to the counter weight. Note the escape velocity locus. Any vehicle launched from the cable beyond this altitude will leave the Martian system without other effort. The plane of revolution is the 61st parallel. For this cable length the cable tension at the counter weight is 7.5 times the force of gravity. This makes the system fairly stiff against outside influences. For a 35,000km cable cable tension is 5.6 times gravity and the end is over the 53rd parallel. For a 30,000km cable the cable tension is 3.2 times gravity and the end is over the 43rd parallel. Shorter cables are cheaper but less stable in the presence of outside influences such as gravity from the moons and varying freight and vehicle transportation loadings. The closer the end gets to the equator, the less stable it is.

Please leave any comments in my comments post.

A windage myth

A common misconception about windage is that the wind speed at the target is the most important. Superficially this seems reasonable since the slower a bullet travels the more it is deflected by the wind. An M855 is deflected by a 10MPH wind by 3.6 inches between 0 and 200 yards, by an additional 4.6 inches between 200 and 400 yards, and by an additional 6.4 inches between 400 and 600 yards which adds up to 14.6 inches but the actual 600 yard deflection is 39.3 inches as shown below. Something else is going on.

The point is that unlike a sailboat, a bullet doesn’t know where the target is. The wind blows it off course and so it is traveling sideways. After the first 200 yards it is traveling in the direction of the green tangent above. If there were no wind between 200 and 600 yards it would continue in a straight line with a total deflection of about 18 inches. The wind for the first 200 yards is responsible for 47% of the deflection. Similarly the wind between 200 and 400 yards is responsible for another 37% and the final 200 yards of wind at the target end is responsible for only 16% of the total windage deflection.

Pay more attention to the wind speed and direction at the shooting end for better results.

Please leave comments using the post in my comments category.

Low Latency Contact Debounce Circuit

Electrical switching contacts are not nearly as simple as they seem.  Both at make (on) and break (off) they produce various undesirable side effects.  Transition times in the nanosecond range can produce unexpected reflections, surges, and arcs in long lines.  As the contacts make and break they bounce microscopically causing rapid on and off pulses lasting up to tens of milliseconds.  Sliding of the deflecting contact surfaces causes fritting that generates even faster pulses in the microsecond range. At circuit potentials higher than a few volts the bounces produce arcs whose effects range from electromagnetic interference to ignition of explosive atmospheres. This applies to light switches, electromechanical relays, electromechanical timers, and indeed any contact between solid conductors at different potentials.

In our modern solid state low voltage world, contact bounce causes problems with switch interfaces to computers and data gathering systems.  If you try to count switch events with an electronic counter it will indicate tens or hundreds of counts for each actuation.  A typical computer can read the state of a switch hundreds or thousands of times during the contact bounce time resulting in multiple “clicks” as far as the computer is concerned.  The normal solution to this problem is some sort of “debounce” circuit that filters out short transitions and eventually settles to the desired state after 50 milliseconds or so.  This works well for mouse buttons or remote TV controls.

Unfortunately this is undesirable for instrumentation, test, or metrology applications, especially multi-channel applications.  The delay time will vary from channel to channel and will be affected by the bounce duty cycle.  This will make it impossible to accurately determine the time of the initial event closer than a few tens of milliseconds.  As an example, in industrial testing it is common to test cam switches at higher than normal speed and correlate the event timing with the cam dwell times which requires very accurate measurements of the initial event timing and sequences.

This circuit essentially solves the contact event measurement problem.  The switch signal passes through a spike filter and then through a transparent latch.  The output of this latch is the cleaned-up input signal and is applied to an exclusive-or gate with a delay on one input.  When a transition passes through the latch, the two inputs of the gate are briefly different.  This causes the output of the gate to pulse low, holding the state of the latch for the duration of the delayed channel of the exclusive-or gate.  This delay is selected to exceed the length of any expected contact bounce but to be shorter than any expected valid signal repetition.  At the end of this time the exclusive-or gate unlatches the transparent latch and it reverts to normal.  Note that since the contact will have completed its event the output of the latch does not change.  As the exclusive-or is not polarity sensitive, this circuit works the same way for rising or falling edges.  One consideration is that the data input to must hold steady until the exclusive-or signal is able to latch it.  This is about two propagation times or a few nanoseconds for typical logic families and is the purpose of the input spike filter.  Its input time constant should be just long enough to guarantee this condition.

This circuit enables contact event measurements with time resolutions of a few nanoseconds.

Please leave comments using the post in my comments category.

Locking Loop Knot

A particularly good knot for tying down tarpaulins is the slipped locking loop.  This is an adjustable self-locking loop knot that is easily untied, even when frozen in winter.  It is described on page 137 of John Shaw’s Directory of Knots.

Tie a stopper knot (either  Ashley’s or a quad stopper) in one end of a rope and pass it through a tarp grommet.  Take the other end through or around the tie down point forming a loop.  Pull on the working end to make the rope taught.  Form the main loop and slip loop, as shown above, near the tie down point.  Snug up the knot enough that it will just barely slide on the standing end.   While pulling on the working end to tighten the rope to the tarp, slide the knot away from the tie down until it snugs hard against the working end.  The main loop will pull the standing end around the slip loop and lock the knot in place.  Load tension from the tarp end will lock it tighter.

To remove the tie simply pull out the  slip loop and the knot will fall apart.  This feature is very useful for applications like covering car windows in the winter where freezing rain and snow-melt-freezes will completely lock up ordinary knots making them impossible to untie.

Please leave comments using the post in my comments category.

Jointer knife adjustment jig

The jointer is the fundamental woodworking power tool for creating flat, square lumber without cup, bow, or twist.  It is the first step in preparing wood for woodworking.  The most critical jointer adjustment is ensuring that the cutter knives are precisely coplanar with the outfeed table.  Otherwise you will not be able to successfully produce lumber that is flat and free of scarfs.  The typical user guide instructions are to place a straightedge on the outfeed table overhanging the knives and then adjust the knives and/or the outfeed table until the knife edge just touches the straightedge across the width of the table, which is fairly imprecise and gets more difficult as your eyes age.  The illustrated jig allows you to easily adjust the alignment to within 0.001″ or so.

This one consists of an inexpensive ($15 at Amazon) dial indicator mounted on an old surface gauge base.  Note the flat tip on the indicator plunger.  Dial indicator plungers come with ball ends which are not suitable for measuring a knife edge.  These ends are replaceable and all use 4-48 threads.  The flat one above is part number 20625A661.  The mounting strut is a rod end blank, 6065K171.  You can save yourself some machining by cutting down a 6066K34.  It is important to get the axis of the plunger square to the surface.  This makes the measurement insensitive to small changes in position.

With the end of the plunger on the same surface as the base, turn the dial bezel until the indicator index (the small black triangle at the bottom of the dial in the top image) matches the dial reading.  The plunger tip is hardened…if you drop it on your cutter edge you may damage the cutter.  Pull up on the top of the plunger and slide plunger over the knife edge.  Then gently lower the plunger.

Follow your jointer cutter/outfeed adjustment instructions to bring the dial reading into alignment with the index flag.  Repeat across the outfeed table.

The base doesn’t need to be this fancy, just stable.  I happened to have this on hand.  A piece of angle iron with three screw adjustable support points to square it up would do as well.

Please leave any comments using the post in my comments category.


The Ortgies is a 1920-1927 production German pocket pistol that was very popular at the time and was imported in large quantities into the US.  It is externally similar to the Colt pocket pistol but completely different internally.  It is comfortable and pleasant to shoot.

In many ways it is an elegant design.  It has no screws.  It is nicely finished with no corners to catch on a pocket.  As a 100 year old striker fired pistol it should never be carried with a round in the chamber.  Here is a very good article about these pistols, and another here.

The main quirk of these pistols concerns the firing pin spring guide rod.  Firstly, when you field strip one of these the guide rod launches itself (with or without the spring) out the back of the pistol.  Field stripping instructions usually include an admonition to cover the back of the slide with a finger, disassemble in a plastic bag, or just to “take care” to not lose the guide rod.  Secondly, assembly is difficult unless you know the “trick”.  There is an assembly notch on the underside of the slide above the guide rod channel.  To assemble, compress the guide rod and spring and push it up into the notch.  Then you can replace the slide normally.  This is best practiced a few times without the firing pin spring and guide rod as bobbling the slide installation will also launch the guide rod out the back.  There are at least two different Ortgies guide rods.  One is 0.842″ long which is too long to fit in the assembly notch of some pistols.  Here is one that works in a fifth style 32ACP pistol which is the majority of production.

Note that it is important to insert this into the small end of the spring and insert the large end of the spring into the firing pin housing.  Otherwise the pistol will jam.

One other note, do not attempt to remove the grip panels without comprehensive disassembly instructions.  There is a latch inside the magazine well which must be pressed in to release the back edge of the grip panels. The front of the panels are hooked into the frame.  Great care must be taken as the wood engagement surfaces are very small and comparatively fragile.  In general, the grips should not be routinely removed as this is unnecessary for cleaning and only necessary for a detail strip.

Please leave any comments using the post in my comments category.


I came across a bit of poetry by Emerson that is apropos to the times we find ourselves in:

Each the herald is who wrote                                                                               His rank, and quartered his own coat.                                                         There is no king nor sovereign state                                                              That can fix a hero’s rate;                                                                                  Each to all is venerable,                                                                            Cap-a-pie invulnerable,                                                                                     Until he write, where all eyes rest,                                                                 Slave or master on his breast.

Fast Settling PLL Multiplier

Phase Lock Loops have long been used for clock recovery, fixed frequency multiplication, reducing clock jitter, frequency synthesis, FM modulation / demodulation, and other tasks.  The basic PLL consists of a Phase/Frequency Detector, loop filter, and Voltage Controlled Oscillator whose output is fed back to the detector through an optional digital frequency divider.  When the loop locks to the input frequency, the output frequency is equal to the input frequency times the divider ratio.

Direct Digital Synthesis circuits have supplanted PLLs in many synthesizer roles but the PLL has the advantages of cost, simplicity, and the ability to maintain constant phase lock to the source.  RCA application note ICAN-6101 describes the use of a CD4046 Phase-Locked-Loop IC as the heart of a 3 digit 1KHz to 1MHz synthesizer.  Note that for synthesizer applications it is necessary to use the Frequency detector PC2 rather than the simple Phase detector, PC1.  The biggest issue with this circuit is that the loop damping factor varies with the square root of the loop gain which is the product of the PFD and VCO gains divided by the division ratio.  The PFD gain is in volts per Hz and the VCO gain is in Hz per volt so the loop gain is dimensionless.  At high division ratios, i.e. high frequencies, the loop is under damped and is over damped at low frequencies.  This is a trace of a standard PLL synthesizer  VCO control input switching between 3KHz and 1.024MHz.

Note that the response at the high frequency end is oscillatory and highly under damped with a settling time of about 200 milliseconds.  The low frequency response is highly over damped with a similar settling time.  The loop filter constants were chosen to equalize settling times.  Improving either degrades the other.  This circuit uses a CD74HC4046 for a faster VCO at 5 volts.  Unlike the CD4046, the HC4046 VCO has a limited common mode range at the VCOin pin and is normally only capable of a  5 to 1 frequency ratio.  To get ratios of 300 to 1 or higher there is a simple workaround.

By grounding the VCOin pin and applying the VCO signal as a sinking current source to the frequency offset input the full range of the VCO is available down to essentially zero Hz.  R6 raises the low frequency voltage enough to avoid Vos problems with U2. R3 and R4 limit the Q1 common mode range to keep it under the pin 12 bias point.

The RCA application note suggests switching in different loop filter components for different frequency ranges but this doesn’t really fix the problem and is less practical with digital control.  Traditional PLL loops have linear VCO gains.  This is essential for applications such as FM modulator / demodulators.   The main insight here is that the settling time is determined by the loop gain at the target voltage.  The gain at other voltages is not relevant.  If, as the division ratio increases, the VCO gain could be increased, the two would cancel and the damping factor would be constant, allowing simple loop compensation.  The solution is an exponential VCO gain response with a low gain at low frequencies (low VCO input voltage) and high gain at high frequencies.

The exponential VCO gain is 4.5KHz per volt at 1KHz and 4.5MHz per volt at 1MHz.  One thousand times higher gain at one thousand times the frequency.  The question arises as to how to make such a VCO.  The obvious solution is to use the Vbe versus Ic relation of silicon junction transistors.

The 470 ohm resistors protect the transistors in case of fault or overload.  The diodes are Schottkys.  R provides a phase lead to the integrator to compensate for the propagation delay of the Schmitt gate at high frequencies.  The matched pair HFA3096 transistors have a gain-bandwidth product of 8GHz which precludes ordinary breadboarding due to parasitic oscillation.  A lower speed design was used for this implementation.

This circuit generated the gain graph above.  Q1 is sufficient for the exponential function.  As this is only needed for loop stabilization the Vbe temperature coefficient is irrelevant.  Note that the loop compensation filter capacitor is one fifth the size of the linear VCO circuit above.  The 3KHz to 1.024MHz step response of this circuit is:

The settling time for both transitions is 900uS and, when zoomed in, both edges are critically damped at high and low frequency.

These kinds of frequency multipliers are useful for synchronous sampling of repetitive signals such as FFT analysis of variable speed rotating equipment at varying resolutions.  While FFTs usually are preceded by a windowing  function to avoid asynchronous artifacts, these functions broaden spectral lines and reduce the spectrum resolution.  With the ability to select FFT sampling rates that are, for instance, 256, 1024, or 4096 times a repetitive signal with a phase locked sampling clock, the requirements for  windowing functions can be reduced or even eliminated.  This enables trading between resolution and data processing overhead for optimized real-time monitoring.

Please leave any comments using the post in my comments category.


An Alternative to Dark Energy

In 1998 when the accelerating expansion of the universe was reported by two groups, the astrophysics community shortly declared the existence of some kind of “dark energy”.  While no one knew what this was, physicists invoked things like Einstein’s Cosmological Constant and vacuum energy to fit it to various theories.  As John von Neumann famously quipped, “With four degrees of freedom I can fit an elephant.”  While there are theories that attribute the acceleration data to local rather than global effects or sampling issues let’s assume the acceleration is real and global.  If so and the acceleration is caused by dark energy / vacuum energy / cosmological constant, the universe may come to a practical end in terms of habitability in 500 billion years or so.  This is a long time and may be an optimistic estimate.

Here’s an alternative to dark energy.   The starting point is the question:  Why are we here — now?   The philosophical name for this is the doomsday argument.  The basic idea is that while the universe is supposed to last for trillions of years fading into a heat death, here we are at the very start, only 13.8 billion years in, only 1% of the first trillion years.  Since the early universe did not generated the elements needed for life, the ratio is even worse.  Being born “now” in the universe seems unlikely.   Locally, even with speed of light limitations, we should be able to spread around our galaxy in no more than a few million years.  We have found out recently that most stars have planets.  Out of the 400 billion star systems in the Milky Way, there are perhaps a billion suitable planets that we could populate.  If that is going to happen, finding ourselves on a single planet out of a possible billion is unlikely.  If you reach into a jar with 10 balls numbered 1 through 10 and pull out number 6, nothing seems odd.  If you reach into a bin with a billion balls, numbered 1 through 1,000,000,000  and pull out number 6, that’s weird.  This is a known problem in statistics with a well defined confidence interval based on how close you are to the start of something.  What it boils down to is that a random sample can be expected to occur between 3 and 97 percent of the range.  If you pull out a 6 the first time you can be 95% confident that there are between 7 and 200 balls total.   After 13.8 billion years we can expect the universe to last from 200 million to 445 billion years from now.  Homo sapiens have been around for 200,000 years, so to be here now, we expect H. sapiens to last another 6 thousand to 6 million years.  The implication is that we may not be around long enough to populate the galaxy and/or the galaxy may not be around as long as we think it will be.  The first possibility involves things like global nuclear/biological war, asteroid strikes, and the like.  The second possibility is the subject of this post.

We are alive at about the earliest time possible in the history of the universe.  Modern scientific discoveries have moved this forward.  The original Milky Way consisted of population 3 stars made up of hydrogen and helium and little else.  Life was not possible.  Some of these stars ended their lives as supernovae creating and ejecting elements like nitrogen, oxygen, calcium, phosphorus, and carbon necessary for life as well as iron and nickel useful for making solid planets.  These enriched the hydrogen clouds that eventually were swept into the formation of population 2 stars, some of which ended as supernovae producing the additional elements that went into the formation of population 1 stars like our sun 4.5 billion years ago which is already 2/3 of the current age of the universe.  So far, so good.  Fifty years ago the assumption was that the solar system was typical and life was inevitable.  The earliest life appeared nearly 4 billion years ago.  It took 3.5 billion more years for anything more advanced than bacteria to appear.  450 million years later we’re here.  Presumably other paths would create self-aware life earlier.

Then modern science complicated the picture.  While we have discovered that most stars have planets, the solar system is atypical and most systems have hot Jupiters and other barriers to the kinds of planets that are needed for life.  In our system, the Jovians, rather than eating the terrestrial planets, protect them by clearing the inner system of asteroids and comets.  Quite unusual and unlikely.

Then there’s the Moon.  The Moon is by far the largest moon compared to the planet it orbits.  It is responsible for tides that may have been necessary to enable the movement of life from the seas to the land.  More importantly it stabilizes the earth’s rotation axis, keeping the seasons regular.  Without it, the earth would slowly tumble, alternating freezing and desert climates, and preventing the emergence of complex life forms.  But the moon is almost impossible dynamically.  As a result of the Apollo sampling missions and modern computer simulations we have a pretty good idea what happened.  At some point a few tens of millions of years after the formation of the solar system a fortuitous Mars sized body slammed into the earth at just the right angle and speed to create a spinning disk of rubble that eventually settled into the earth and the orbiting Moon … an extraordinarily unlikely event.

Uranium is the next issue.  The radioactive uranium in the earth’s interior has kept the iron/nickel core molten for the last 4.5 billion years.  This allows the core and mantle to produce a magnetic field.  This field shields the the earth and the life on it from the solar wind and solar flares.  Without it the earth would be a radiation seared wasteland like Mars and the Moon.  The problem is, where did the uranium come from?  We know it was not part of the early universe.  And it turns out, the supernovae that produce lighter elements are not able to produce uranium.  No uranium, no life.  Enter LIGO, the gravity wave observatory.  Analysis of the gravity waves yields a possibility.  Most of the “visible” events are black hole mergers that yield no material output other than gravity waves.  Similarly black hole / neutron star mergers only produce gravity waves.  A few of these events are mergers of orbiting neutron stars.  These produce an outpouring of large nucleus elements including uranium in addition to gravity waves.  While in the minority, we finally have a source for uranium, sort of.  To have a merger of neutron stars we have to start with two orbiting stars both between 10 and 29 times the size of the sun, no more, no less.  Smaller stars turn into white dwarfs without the density to create uranium in a collision and larger stars turn into black holes that do not release anything.  These two stars have to live out their lives, both supernovae without disrupting the other producing orbiting neutron stars.  These then slowly spiral in and merge and explode.  This takes a long time and is quite rare.  This has to happen near a population 1 star forming region to seed the gas clouds with enough uranium for life friendly planets. This would be relatively nearby and 5-6 billion years ago for us.  LIGO sees these events from all over the universe every few weeks, but since there are about a trillion galaxies in the universe,  there may only have been one in the our entire Milky Way galaxy prior to the formation of the solar system.  This tightens the constraints even more and indicates that we are really living at the earliest time possible , which makes a long future for the universe even more unlikely.

Consensus in the astrophysics community is that our universe is a finite unbounded space, probably a 3-sphere embedded in 4 space, just like our planet is a 2-sphere embedded in 3 space.  A recent astronomical geometrical finding indicated that we are indeed in a 3-sphere.  As you can travel on the earth’s surface in 2 dimensions forever without reaching an edge, you can travel in space in 3 dimensions forever without reaching an edge.  Our 3-sphere universe is probably embedded in a 4-sphere and so on.  This hierarchy sounds like an infinite universe but not really.  As it turns out the volume for an N-sphere of a given radius only increases up to the 5-Sphere and then starts decreasing rapidly and the infinite sum converges.  If you add up the volumes of all the hyper-spheres of unit radius (say one universe radius) the total volume is 45.99932…, ie. the total of volume all the finite universes is finite.  (Yes, it’s turtles all the way down but they get very tiny quickly.)  I’m ignoring the various infinite multiverse theories as these appear to be string theorists grasping at straws.  A 1-sphere on a 2- sphere is a circle like a fairy ring of mushrooms on the surface of the Earth.  The ring has a finite interior that takes up a finite fraction of the surface of the earth’s 2-sphere surface.  The ring has a center which is part of the 2-sphere but not on the ring.  Earth’s surface 2-sphere has a finite volume that takes up a finite fraction of the 3-sphere we live in.  It has a center that is in the 3-sphere but not on the surface.  Similarly, our 3-sphere 3 dimensional universe has a finite volume that tales up a finite fraction of the 4-sphere it’s embedded in.  It has a center in the 4-sphere that is not in the 3 dimensional 3-sphere universe.  That center probably shares 4 dimensional coordinates with the center of mass of our universe and the historic location of the big bang in four space.  Our expanding 3 sphere universe spreads through the 4-sphere just like a fairy ring spreads over the surface of the earth.  The main point here is that if the total multidimensional universe is eternal and “big bangs” happen on a regular basis, the finite 4-sphere will have already filled up with expanding 3-sphere universes which, if they persist, will be banging into each other.

As scientists were searching (successfully) for the hypothesized Higgs Boson, vacuum decay entered the discussion.   This represents the possibility of a point quantum event that disassembles and destroys everything the the universe, with the wave front traveling at the speed of light.  While this is considered unlikely, the science is  not settled.  There is also the possibility that a collision of 3-spheres could trigger such an event.  We may live in a soap bubble waiting to be popped.  Since most of the universe is beyond our light speed horizon it is possible for such an event to occur without ever reaching us through three space.  It is conceivable that shock waves propagating through 4 space are faster than the 3 space speed of light, much like seismic P waves propagating through the earth faster than the surface S waves.  While the decay event propagates through the 3-sphere, the shock wave could couple to 4 space and travel across the 3-sphere interior much faster.

Perhaps 5 billion years ago a decay event occurred somewhere in our universe.  The initial decay volume produced a pressure shock that propagated in 4 space across the 3-sphere internal volume and inflated the universe slightly.  As the surface of the volume of destruction increased, the generated shock increased which increased the inflation of the rest of the 3-sphere.  Thus the existing expansion of the universe appears to accelerate.  If this event occurred within our horizon, that may explain why we aren’t here much later.

On a happier note, assume the bubble doesn’t ever pop.  Two popular scenarios in that case are the Big Rip and Heat Death of the Universe.  Both involve runaway expansion leading to an effectively empty, infinite universe.  Really boring.  Consider the fairy ring — when it’s a few feet in diameter we can imagine it expanding forever — but we know that if it did, it would crunch back together on the other side of the earth (ignoring oceans and deserts).  Similarly any Big Rip or Heat Death will eventually meet itself on the other side of the 4-sphere.  Who knows, it may start another big bang.

Please leave any comments using the post in my comments category.