Sunday, October 26, 2014

Inertia

NOTE: The website physicsclassroom.com is a fantastic resource for physics teachers and students! It does an especially good job with forces. There are several links to it in this article.

The concept of inertia seems simple. We learn that an object in motion tends to stay in motion and an object at rest tends to stay at rest. This is Newton's first law of motion. Dominique Toussaint;Wikipedia
We experience inertia from the moment we start playing with toys as toddlers. Not surprisingly it is a concept that usually makes intuitive sense when we encounter it more formally in junior high school. So what's there to explore? As with many concepts in physics, inertia makes prefect common sense as long as we stick to the classical definition and avoid poking at it. But how does it work in Einstein's space-time? And have you ever asked yourself why objects have inertia? Why does an object resist any change in motion? The picture grows more mysterious still when we try to put inertia on a cosmological scale.

The Basics

Let's start by refreshing our understanding of inertia. It is common to say that inertia is an object's resistance to any change in its velocity, whether that velocity is zero or close to light speed, and in most everyday cases this definition works perfectly. We can also think of inertia as the tendency of an object to maintain its current momentum, which is written as mass x velocity.

It is important to note that inertia is not a force. A force is any interaction that causes an object to change velocity, which can mean putting a stationary object into motion or bringing a moving object to rest, or it can mean changing the direction of an object's motion. In any case, the object experiences acceleration. Force = mass x acceleration. Neither momentum nor inertia is considered a force for this reason. This concept can get a bit tricky because it is tempting to consider inertia as a kind of countering force - the inertia of an object seems to resist or "push back" against any change in motion (as the result of a force that is applied to it). However, inertia is the tendency of the object to resist any change in motion, not the resistive force (such as friction).

Inertia is not written as an equation and there are no units of inertia. It is simply the tendency of an object to resist any change in motion. Like momentum and force, the inertia of an object is directly proportional to its mass. Unlike momentum and force, inertia does not depend on an object's velocity, not even a little bit (at least in classical Newtonian mechanics!).

Evidence Of Inertia

How do we observe inertia in action? We can single it out by eliminating two phenomena that are commonly confused with inertia - friction and the force of gravity. Imagine throwing a ball straight up in the air. The ball will resist this change in state - it will resist the change from being at rest to being in motion. We have gravity at work on the ball, however, and we want to eliminate acceleration due to the force of gravity, so to simplify this scenario, we will imagine instead rolling the two balls on a floor, each one launched with the same force of your hand. Now that we have ruled out gravity (for the most part!), we will see the heavier ball come to a stop much sooner than the lighter ball. Or, we could say that we need more force to get that heavy ball to roll as far as the light one. Both ball exhibit inertia: They resist being put into motion (a launch force is needed) and they resist stopping once they are in motion (the force of friction is needed).

We quickly realize that mass has everything to do with inertia.

We start to wonder: Did the heavier ball simply experience more friction in our ball experiment? Friction is often confused with inertia. Friction is the force that resists the motion of objects, and it is a contact force. Although friction is a complex phenomenon, it can be quite accurately modeled and the mechanisms behind it are well known. We can eliminate friction by repeating our experiment in the vacuum of space (where no contact forces are present). We can also (mostly) eliminate any gravitational influence by performing it far away from any other objects as well. The lighter ball will go faster than the heavier ball if both are launched with the same force, but they will not stop as they did on the floor. Both balls will keep on going off into space at constant velocity until we can't see either of them anymore, unless a force acts on them. Without friction at play, the presence of inertia stands out: once an object is in motion it tends to stay in motion.

The same force is applied to each ball but the heavier ball has more mass so it will not accelerate as much as the lighter ball (force = mass x acceleration). The heavier ball will not attain as much velocity in this force-applying collision as the lighter ball will, but its mass is greater. Both balls will have equal momentum.

Testing how two objects of different masses behave in an environment that offers no outside influence - no source of friction, no gravity - seems to make it clear: Inertia can be boiled down to something inherent in objects, and that something seems to be mass. In our experiment only mass makes the motion different between the two balls.

Mass and Inertia

So what about mass? I offer first a direct quote from Wikipedia that I think is very apt: mass is " a property of a physical body, which determines the body's resistance to being accelerated by a force and the strength of its mutual gravitational attraction with other bodies." If we set aside the gravitational reference for the moment, we essentially have a definition of inertial mass.

We now have a definition of inertial mass but what about the relationship between gravity and mass? We've been confidently talking about our lighter ball and heavier ball. Let's assume we've weighed them carefully: 150 grams and 250 grams respectively. We can infer then that 250 grams has more inertia than 150 grams, which is supported by our observations. But as we'll see, we've actually measured the ball's weights not their masses. If we took the balls and the scale to the Moon, we would find they weigh only about 42 grams and 30 grams, respectively. What we are measuring is gravitational mass. Newton realized that what we are measuring with our scale is actually force rather than mass. A painted portrait of him is shown below right. Handsome! Isaac Newton
It is a force equivalent to mass x acceleration where acceleration is the result of the mass in question being inside a gravitational force field. At the surface of Earth, all objects experience a gravitational acceleration of about 9.78 m/s2. Force measured in newtons = mass x acceleration. On the Moon's surface the gravitational acceleration is much smaller, only 1.62 m/s2. Why this difference? Because Earth has much more mass, it exerts more gravitational force than the Moon does. Likewise, our larger ball exerts about 60% more gravitational force than the small one does. In this case, these forces are so small compared to Earth's overwhelming gravity that we don't notice them. The Cavendish experiment (1798), using a sensitive torsion balance, was the first measurement and confirmation of the gravitational fields around (smaller) objects on Earth.

If we put the balls in the International Space Station the balls will float around the cabin because objects experience no gravitational force there. Yet - the cabin still experiences the gravitational force of Earth! It's not all that far away from Earth's surface. It is much closer than the Moon, for example, which is held in orbit by Earth's gravitational field. The reason for this is that the station is placed in precisely the right orbit and maintains the right orbital velocity to balance the (forward) inertia of the station, which is its tendency to fly off into space, with the gravitational force (which makes it fall to Earth).

This inertia is sometimes called centrifugal force but, strictly speaking, this is incorrect. Centrifugal force is actually a fictitious force rather than a real force, something we will explore further in this article. This can be confusing because, according to Newtonian mechanics, the inertia we are talking about here is an equal and opposite reaction to the centripetal force, which is a real force.

The Space Station is in fact falling to Earth but its free fall is curved because it is traveling fast and that curve matches the curvature of Earth so it maintains its altitude. Gravitational force in this case acts as the centripetal force balancing the "centrifugal force" or, stated more accurately, the inertia. For example, if we twirl a ball attached to a string we are holding, the tautness of the string that keeps the ball in orbit is centripetal force. The ball wants to fly off because it has inertia. Once in motion it tends to stay in motion. Because it is rotating, the direction of its velocity is constantly changing, and it is therefore experiencing acceleration. If Earth suddenly disappeared, the ISS and the Moon would fly off in the direction tangential to their precise positions at that moment, just as the ball on the string would do if you let go of the string. Their inertia at that moment would be forward in the same direction as their velocity at that moment.

What about an object far away from any gravitational field? It will weigh nothing so what is its mass? The mass is still the same, just as the masses of the two balls remains constant no matter what planet we weigh them on. Their masses are about 15 and 25 grams. We could say that mass, in its most naked sense, is the amount of matter in an object, the numbers and sizes of the atoms it contains, in other words. However, even this definition is complicated, as we will see in a moment.

We've untangled inertia from mass, friction and gravity, but we still have two more scenarios to figure out: first, inertia and its mass-energy equivalent and second, mass/energy (and therefore inertia) in different frames of reference. These concepts are a bit more challenging.

Mass and Energy are Equivalent

If mass and energy are truly equivalent then energy too should experience inertia, and it does. I said previously that it seems that the mass of an object should simply be the sum of the masses of all the protons, neutrons and electrons of the atoms it contains. It turns out that we can't do this, thanks to Albert Einstein's mass-energy equivalence (His portrait is shown below). Albert Einstein
There are two things to consider in terms of mass-energy equivalence.

The first is mass defect. An atom is a bound system. It has lower energy than the summed up energies of its particles - neutrons, protons and electrons. Because its bound energy is lower, an atom has less mass than the added up mass of its unbound particles. This means that when an atom binds together, some of the mass of its particles turns into energy and that energy is lost to electromagnetic radiation (heat and light). This doesn't mean that a bit of each particle is shaved off and disappears. It means that the particles themselves assume lower energy states in the atom, because each particle's mass is partly an energy contribution.

When an atom is in a very excited state with high kinetic energy it has more mass and it must be cooled to a state at rest before mass defect shows up and can be measured as mass defect.

The second thing to consider is relativistic mass, or mass traveling close to the speed of light. Einstein's theory of special relativity deals with these kinds of scenarios. All objects have an invariant or rest mass (the ball's masses of 15 and 25 grams are examples), which is the lowest possible mass, and it is the same no matter where or how you measure it. However, objects in motion relative to you have an extra contribution to their mass/energy in the form of kinetic energy. The contribution of kinetic energy becomes more significant as the object approaches light speed, thanks to the nature of Einstein's famous equation, energy = mc2, where c is the speed of light. If you measured the mass of an object flying past you at near light speed, it would be greater than its rest mass. If you decided to accelerate yourself and then travel along with that object, its measured mass would now be its rest mass. Relativistic mass increases but the substance of the object, its rest or invariant mass, remains the same.

What about inertial mass? Now we have a departure from classical mechanics. According to classical mechanics, inertia does not depend on velocity, but at velocities approaching light speed inertial mass increases. In fact, inertial mass and relativistic mass are equivalent measures. Now we can say that there is a preferred value for inertial mass and that is its mass at rest relative to the measurer. This even holds up for massless objects such as photons. A photon's preferred inertial mass is zero; it has no rest mass. However, a photon, like an object with mass, has both inertia and momentum (momentum is now more correctly thought of as mass/energy x velocity). Even though the speed of a photon is constant no matter how and where you measure it, whether you are moving or at rest yourself, its inertial mass will vary depending on your motion relative to it. Measuring the object's inertial mass, as well as its momentum, now depends on from what point you are measuring it (are you at rest or are you moving along with it?). It depends on which frame of reference you use. What is a frame of reference and why is it important? This 7-minute video by Derek Owens answers both questions very simply.

Inertia And Frames of Reference

When Isaac Newton worked inertia into his laws of motion in the late 17th century, a frame of reference was needed in order to describe how it worked in time and space. He viewed this reference frame as one in which his first law of motion, the law of inertia, is valid. It is therefore called an inertial frame of reference. He decided that this could be obtained by using a reference frame that is either in uniform motion or stationary relative to the fixed stars. This means it is neither rotating (a form of acceleration as we just learned) nor accelerating relative to the "fixed" stars. You might be wondering: are the stars fixed? Keep this question in mind as we continue.

Inertial Reference Frame 1. Galilean Transformation

Newton's space, as he understood it, is homogenous or uniform in all directions and it operates independently from time. To understand how his frame of reference works, you can take measurements, such as the velocities of the balls we were talking about earlier, in one frame of reference, such as looking at them going past you, and transform their motion quite easily into another frame of reference in which you are looking directly at them coming toward you. Then, you can describe the motion in yet another frame of reference, such as one in which you are travelling alongside one of the balls, and so on. These transformations are called  Galilean transformations in Newtonian physics. The graph below left illustrates how such a transformation can be described mathematically, by the addition and subtraction of vectors. There are two coordinate systems, where the xyz coordinate system is transformed by vector addition to the x'y'z' coordinate system.

Physical laws, such as Newton's laws of motion, work the same way in any of these inertial frames, as they are called.

This kind of transformation assumes that time is universal and independent and the motion of all objects involved is uniform (vector V (red) doesn't change in magnitude or direction).

Inertial frames of reference are important when we talk about any kind of motion, and in the early 1900's, Albert Einstein utilized Newton's inertial frames of reference when he formulated special relativity. Einstein realized that it is impossible to know whether an object is in motion or not without some outside frame of reference to compare it to, and solving this problem formed the basis of his theory of special relativity.

Inertial Reference Frame 2: Lorentz Transformation

Special relativity works beautifully to describe how light works in space and time. Einstein incorporated the then-groundbreaking fact that the speed of light is a constant and because it is, time and space are not (time dilation and length contraction occur). This was a huge departure from Newtonian physics, where time and space are both constants. Special relativity also incorporates the important realization that mass and energy are interchangeable.

In order to describe how space and time vary with relative velocity, special relativity is constructed on a Lorentz frame of reference, which, like Newton's Galilean frame of reference, is a type of inertial reference frame. The animation below, done by udiprod, is a visualization of how such a transformation works. You'll see that, in order to preserve light speed, everything else must change.

In fact, a Lorentz transformation contracts into a Galilean transformation when velocity is not close to the speed of light. Incidentally, Einstein treats the speed of light not as a property defined by light itself but instead as a property built into space-time: not only light is limited to light speed but everything else as well. This theory (in its basic form!), however, has a bit of a weakness. It has in it the same limitation that Newton had: you cannot describe or transform a phenomenon that involves acceleration. The transforming frames must be in constant motion relative to one another. This is the limitation of using inertial frames of reference.

General Relativity Is a Non-inertial Frame

There is an assumption required for Newton's first law (the law of inertia) to work. The straight-line motion of an object assumes that it experiences a zero net force. In reality, however, any object, even one located very distant from any other objects in space, is acted on, at least somewhat, by the gravitational attraction of other bodies, meaning that there is no practical way to realize Newton's first law. As Newton defines gravity, it is a force that accelerates masses so it cannot be dealt with using inertial frames of reference. This problem led Einstein to develop general relativity, a new theory of space-time and gravity.

Newton's gravity is an attractive force that accelerates masses toward one another, and this means that you must deal with frames accelerating with respect to one another if you want to describe gravity in space-time. You must turn to a non-inertial frame of reference. This introduces two complicating factors.

Consequence 1: Laws of Physics Don't Hold Up

First, out of necessity, the laws of physics no longer hold up in one frame relative to another one. Non-inertial frames of reference violate both the law of inertia (hence the name non-inertial) and Newton's laws of motion. Within an accelerating frame, for an example, an object can change its velocity when no apparent net force is acting on it. To see how this works, imagine you are driving in a car and you've set up a sport cam to video yourself as you drive. Now you apply the brakes hard. You can feel your body move forward and strain against your seatbelt; you can feel a force pushing you forward. When you take at look at your video footage, you wonder where is this forward force suddenly come from? From within the car there is no culprit to be found. In this reference frame, Newton's laws of motion just got violated! The car is an accelerating (or decelerating) non-inertial frame of reference. Meanwhile, an observer standing on the street sees you pass by, and he sees the laws of motion working just fine. He sees a typical example of Newton's first law of motion at play: your body continues to move forward because it tends to maintain its forward motion; it has inertia. The car is stopping but your body resists the change in motion. He can easily see that you have not been spontaneously pushed forward. But inside the car, it looks and feels to you like you have. This apparent forward force is not a real force.

Consequence 2: Fictitious Forces

The second complicating factor is you have to supplement real forces at play with fictitious forces. That apparent forward force in the example above is a fictitious force.

The Coriolis effect is a fictitious force that often gives students trouble (and teachers a real headache). Imagine that you allow a ball to drop to the ground from a tower. While that ball is falling the Earth is rotating, so there should be an additional horizontal component to its trajectory, right? Several sources online cite this as an example of the Coriolis effect, and they are absolutely incorrect. If you drop the ball you will find that the ball drops exactly beneath its starting position, why? That's a good thing actually. Imagine what would happen when a stewardess pours you coffee on an airplane - the coffee would shoot out of her carafe to the back of the plane at hundreds of kilometres per hour! In both cases, the object that is released maintains the same momentum as the frame of reference it is in. That momentum is conserved, so the ball maintains the same horizontal motion as the Earth as it falls (ignoring Earth's curvature) and the coffee keeps the same forward motion as the plane. From the ground, you would see the coffee falling into my cup and flying past you at several hundred km/h all the while as it falls, if the plane was transparent. But what if, let's say, an alien who maintains a stationary position with respect to the Sun ("he" can see Earth rotating beneath him) drops a package down to you. It will have to be released to the east of you in order to reach you, taking into account the Earth's rotation. This is not due to the Coriolis effect, however. It is due only to the transformation of a non-inertial (rotating and therefore accelerating) reference frame.

You cannot observe the Earth's rotation when you drop a ball on Earth but you can observe it's effect on any object undergoing long-range horizontal motion across Earth's surface. Airplanes, ships and even long-range snipers must take into account the Coriolis effect when they calculate their trajectories. The Coriolis effect is all about deflection and the curvature of the Earth. It acts in a direction that is perpendicular to both the rotational axis and the velocity of the object in question, and the effect is proportional to the velocity of the object. The Earth's rotation causes horizontal motion to be deflected to the right in the northern hemisphere and to the left in the southern hemisphere. It appears that the Coriolis effect is a force acting to deflect the motion of objects but, as before, it only appears this way because we are dealing with a non-inertial frame of reference.

The reason for this deflection has everything to do with the fact that Earth is a sphere (a curved surface) rotating on a north-south axis, so different latitudes experience different rotational velocities. There would be no Coriolis effect if Earth were a giant revolving cylinder (except on the ends). The Coriolis effect is zero right at the equator, where the curvature of the surface is balanced. This effect is actually quite subtle, it only becomes significant over long trajectories. You can rest easy that the Coriolis effect on your golf swing, for example, is only on the order of a few micrometers (but you can certainly go ahead and blame it on the Coriolis effect around your less scientifically savvy buddies).

The following 6-minute video does a great job explaining the Coriolis effect.

We've encountered three kinds of fictitious force so far: Coriolis effect, centrifugal "force" and rectilinear acceleration (the example of you in the decelerating car). They all result from a non-inertial frame of reference whether it is accelerating in a straight line or rotating.

Space-time Curvature

Einstein realized that the force you feel as the weight of your body, and represented as an acceleration of 9.82 m/sor 1 g, is equivalent to the force you would experience being inside a spaceship that is accelerating at 1 g. This is the equivalence principle. Einstein also realized that situations where gravitational force is present are actually special cases of inertial motion. In Newton's physics, objects at constant velocity move in straight lines. In Einstein's physics, objects, such as photons of light, move in as straight a line as they can.

Newton's first law is still there. The difference now is that the lines themselves bend - space-time takes on a curvature and becomes what is called a geodesic instead of the flat worldline of Newtonian physics and the flat (Minkowski) space-time of special relativity. You can think of a geodesic as similar to a map of the world that is stretched and curved around a globe. In general relativity an object in free fall is actually a case of inertial motion rather than an object being acted on by the force of gravity. Gravity is no longer a Newtonian force but a geometry instead, as shown in the following 1-minute animation by Regev. G..

This explains why you would not feel any force acting on your body when you are skydiving in free fall. An accelerometer attached to your body would not record any acceleration. Meanwhile, an observer on the ground watching you fall would see your body accelerate toward the ground until you open your chute.

Einstein reconfigured the flat space-time of special relativity to allow it to curve and stretch. Gravity is a curvature in space-time rather than a force. In this formulation, the principle of inertia is synonymous with geodesic motion. Recall the photon that maintains a straight-line trajectory following Newton's principle of inertia. It has been experimentally shown that the trajectories of photons traveling from distant supernovae do indeed bend toward gravitationally massive objects such as galaxies. The path of the photon itself doesn't bend. It is the space-time it is traveling through that bends, and the photon is simply following its geodesic, its shortest possible path in other words.

This means that in general relativity there is no universal (homogenous or uniform in all directions) inertial frame of reference possible. The curved space-time of general relativity reduces down to flat Minkowski space-time if we are dealing with relatively small regions of space-time. However, even our night sky is slightly distorted due to the gravitational deflection of light caused by the Sun, with the exception of the region of night sky that is directly opposite the Sun where the deflection angle is minimal to zero.

Space-time in both special relativity and general relativity is constructed as a four-dimensional manifold consisting of three space dimensions and one time dimension. However, general relativity as a geometric construction not only describes space-time itself but also the energy-momentum (energy fields and matter) contained within it. On top of that, it is also a Riemann tensor, which can curve, allowing general relativity space-time to bend and stretch, something special relativity space-time can't do. General relativity describes gravitational time dilation, gravitational lensing (the bending of photons around massive objects) and gravitational redshift, all phenomena that have been experimentally verified. Because special relativity cannot describe these large-scale phenomena where space-time curves or stretches out, it must be considered a local theory rather than a universal one.

The End Of A Fixed Universal Frame of Reference

Newton's concept of absolute space, where we have a frame of reference that is stationary relative to distant (fixed) stars, no longer holds thanks not only to general relativity as described above but also to star movement. We now know that distant stars are not fixed at all, even though it appears that way - a constellation appears the same over thousands of years. There are two reasons they are not fixed. First, the Earth moves relative to them just a tiny bit as we rotate around the Sun. This is called apparent motion because Earth is the one doing the moving. However, the star itself also moves (real or proper motion) as it is in a galaxy that rotates and the star rotates with it. As well, the star also moves within the galaxy (called peculiar velocity). Our Sun, for example, dips up and down through the plane of the Milky Way as it rotates along with its neighbor stars within the rotating Milky Way. It moves around in its local star neighbourhood.

Star movement and general relativity are not the only reasons that the distant stars cannot be used as a fixed frame of reference. The universe itself is also undergoing constant change. It is in a state of accelerating expansion. This does not mean that distant stars themselves are moving away from us faster and faster in all directions (although it appears that way). Instead it is the fabric of space-time itself in which the stars are imbedded that is stretching out in all directions at an accelerating rate. This delivers yet another blow to any notion of a universal fixed frame of reference. Even so, in some applications where an approximate frame of reference is required, the distant stars serve quite well because the discrepancy that stems from these effects is quite small.

Other inertial applications require an approxiamte reference that is even simpler. For example, onboard inertial navigation or guidance systems (INS) use a computer, accelerometers and gyroscopes (rotation sensors) to calculate the position, orientation and velocity of aircraft, ships and spacecraft. These systems operate on the basis of inertial space, which is similar to Newton's absolute space except that distant star locations are not required. Their operation has just two requirements - objects experience constant motion relative to one another and they appear to be at rest relative to one another - both satisfied within the local environment of the instruments themselves. All of the instruments in an INS operate on the same process where the vessel frame of reference is compared against the geographic frame of reference. The vessel must usually first remain stationary while it initializes, a process called alignment. External input from additional sensors is usually used to align the system. However, once initialized, the INS offers the advantage of not requiring any external input so it cannot be jammed, and some systems are able to recalibrate while in motion using GPS locators. However, this frame of reference is not perfectly stable; drift inevitably occurs and small errors build as the device input is an iterative process, where one input relies on the previous input.

Now that we've explored what happens to our frame of reference in Newtonian fixed space, in the Minkowski space-time of special relativity and in the curved space-time of general relativity, we see that our concept of inertia absolutely depends on which conceptual environment we put it in, and that conceptual environment can get very complicated! Star positions can approximate a frame of reference and even the basics of Newtons laws themselves can approximate a frame of reference, but is a true universal frame of reference for inertia even possible?

Using Space-time as a Universal Frame of Reference

Recall the ball experiments described earlier. As always, we must define the ball's change in motion with respect to some frame of reference. Inertia requires that our object is moving (or not) with respect to something. Otherwise motion itself has no meaning because we can't distinguish a stationary object from a moving one. We now know that Newton's distant fixed stars do not work as an absolute universal frame of reference with which to judge our changes in motion. No stars or other objects in the universe are perfectly stationary and the universe itself is in a constant state of change because it is expanding.

However, there may be some hope in determining a universal frame of reference. WMAP (Wilkinson Anisotropy Probe) data shows that the universe's expansion has been very smooth in all directions with only tiny local fluctuations. As well, even though gravity curves space-time, overall the universe is very flat, with a curvature of just 0.4%. Furthermore, even though the size of the universe as a whole is unknown, the size of the universe that is observable from Earth is known and has a uniform radius of 46 billion light years. The universe also has a very even density, meaning that although matter in the universe is clumped into galaxy clusters and gas clouds, overall it (and the greater contribution of density by dark matter) is very uniformly spread out. All of this implies that space-time is very much the same here as it is elsewhere in the universe. It is fairly consistent, even in the way it is expanding, an essential quality for a universal frame of reference.

This being said, there may an additional complication faced when looking for a universal frame of reference because there is some evidence that the entire universe may be spinning. There is a slight excess (around 7%) of left-handed spiral galaxies over right-handed spiral galaxies according to the recent Sloan Digital Sky Survey, suggesting a universal preference for spin direction. This data centers on the Northern hemisphere of the night sky so it will be interesting to see if there is a similar excess in the Southern hemisphere as well. If the universe is spinning, then using it as a frame of reference for inertia is complicated because the universe itself will have significant angular momentum. It also asks the haunting question: what even larger frame of reference is the universe spinning within?

Origin of Inertia: What Is the Mechanism?

No one knows what the source of inertia is. It's not satisfying to say that it just is. What exactly IS resisting the change in motion of a mass? Is it a quality intrinsic to matter/energy itself, or is something about space-time at the heart of inertia? Does inertia arise from an interaction between the two?

My previous four articles tagged Quantum Mechanics question whether we can consider space-time as some kind of medium than influences particle behaviour, rather than as an entirely empty vacuum. Most physicists do not think of space-time as a physically real medium of any kind but some interpretations of the very enigmatic Young's double slit experiment, for example, rely on the assumption that the particle exhibits its strange behaviour as a result of interacting with some kind of space-time medium (the pilot or guiding wave interpretation is one). The question of a medium finds yet a different interpretation in fractal theory, explored in my Fractal Universe series of articles. Is the vacuum of space truly empty? Is there an inertial mechanism hidden within this vacuum and can it be used as a universal frame of reference?

What does it mean to say that space-time is a physically real medium? Imagine repeating our earlier ball experiment but instead of launching them into the nearly perfect vacuum of space, we launch them through a vat of maple syrup, something we can agree on as physically real. Now we know exactly where most the resistance to change in motion is coming from - from friction thanks to the viscosity of the syrup, and the friction increases if the balls are not perfectly smooth. If we could get a huge vat of superfluid helium, on the other hand, and float that vat in space, we'd see the balls move through it just as they would move in outer space because we've removed the source of friction (superfluids have zero viscosity or no internal friction in other words). We have a physical medium but it has no ordinary fluid properties. This is the basic idea behind Superfluid vacuum theory, worth a read as the author(s) did a great job of describing how it works in relativistic systems as well as how Higgs field arises in it. Here, we would see once again that the balls resist their change in motion, a resistance owed to their inertia. Maple syrup and superfluid helium are examples of a medium through which an object moves. What if the vacuum of space-time itself is a kind of medium?

Using The Quantum Vacuum as a Mechanism

Some researchers have attempted to define the universal frame of reference as the quantum vacuum of space-time itself. It is widely accepted that, according to quantum field theory, even a perfect vacuum is filled with virtual particles constantly popping in and out of existence, called vacuum fluctuations. There is energy associated with this vacuum. However, there is a catch to this in that most physicists do not consider vacuum energy to be in a useable (thermodynamically accessible) form. Even so, it is possible that inertia acts is a very special form of friction within this vacuum, if particles interact with it as they move through it.

The quantum scale is far too small to observe in any way but what would the quantum vacuum look like when an object moves through it? Is there a quantized wake? Are there ripples? Does it act like water? Does a particle displace the vacuum like the ball displaces maple syrup or does the quantum vacuum permeate all virtual and real particles? We do not have a quantum-scale theory of space-time to give us clues to these questions because we can't rely on Einstein's geometric description of space-time to answer them. It mathematically treats space-time as perfectly smooth on the smallest scale, thanks to his use of partial differential equations. Space-time in his theory bends and stretches, but it does so only on the large (macroscopic) scale. It isn't equipped to make predictions about quantum-sized space-time ripples.

Still it is possible to explore theoretical possibilities. Some researchers such as physicists Bernard Haisch and Alfonso Rueda expect, just as you would with friction, that any change in movement through the quantum vacuum must be associated with the radiation of heat, a real effect that in theory could be verified.

Evidence of this heat would support the theory (and it might mean that vacuum energy can be made thermodynamically accessible) but there is no definitive evidence for it as yet. To further complicate matters, the theory of vacuum energy itself is not without controversy. Does this vacuum have mass and gravitational force associated with it, and if so how does that contribute to the cosmological constant and therefore, the rate of expansion of the universe? The magnitude of the universe's vacuum energy is also very difficult to quantify - how do you measure it except in a relative sense?

This possible vacuum energy mechanism for inertia is based on temperature and an accelerating (non-inertial) frame of reference. It is theoretically well established that in Minkowski (special relativity) space-time an accelerating observer sees the vacuum of space as being bathed in thermal radiation. In other words, it is seen as having a temperature that is proportional to the observer's acceleration. This is the called the Unruh effect. It is an established theory but there is, at the moment, some controversy about whether this radiation has been observed experimentally. In general, this effect relies on quantum field theory: that the vacuum of space is filled with quantized energy fields, rather than simply being empty. The vacuum therefore has a lowest possible energy associated with it that depends on your non-inertial reference point.

For a technical explanation of how the Unruh effect works I turn you to this link in the Wikipedia article. I will attempt to describe it as I understand it: You have two perspectives - one is inertial and one is non-inertial (accelerating). In the inertial (Minkowski) frame you can solve the Schrodinger wave equation and separate your solutions to it into positive and negative frequency solutions that define the vacuum as well as its ("Minkowski") particles. You then solve the equation for accelerating (called Rindler) frame coordinates, and then separate positive and negative frequencies like you did before. When you plug these solutions into the Minkowski frame you get a vacuum that is in a thermal bath of "Rindler" particles, or virtual photons. Doing this, however, means that the concept of a particle is observer-dependent, an insight in itself with possibly far-reaching implications.

What you get for this work is a description of inertial motion that is attached to thermal radiation, and which is different from the thermal radiation associated with accelerated motion. Examples of this latter kind of radiation are Bremsstrahlung radiation and synchrotron radiation. This means that you have a potential mechanism and a way to experimentally test for and quantify inertia.

An interesting aspect to this is that the Unruh radiation would have to be restricted to certain wavelengths because the vacuum energy arising from virtual particles must necessarily be quantized energy. Also, the slower an object's acceleration is, the longer the Unruh wavelengths should be. Eventually they become as long as the observable universe is wide, as all the while more and more wavelengths are disallowed. As a consequence of this, an object's inertial mass increases as its acceleration increases - you get more Unruh photons that contribute to its inertial mass. The object would then be expected to decelerate as it gains inertia. There should be a balanced state achieved where an object reaches a minimal acceleration rate that should approach the cosmic acceleration of the universe. What makes this theory powerful is that it predicts observed galaxy rotation rates, which are significantly higher than the galaxy's visible matter predicts, without invoking dark matter. What weakens it, however, is that it violates Einstein's well-established equivalence principle described earlier: that gravitational (visible matter) mass and inertial mass (now being visible matter + Unruh radiation) should be equivalent. The equivalence principle has been thoroughly tested. Laser ranging of the Moon to measure how fast it falls around Earth proves that the equivalence principle is accurate to within a few parts per trillion.

Setting aside this big problem, we can consider a way to test this theory without having to observe Unruh radiation directly. If we think back to Galileo's famous experiment where he drops two objects (of similar size and shape) off a tower, one being heavier than the other, both objects fall at the same rate, around 9.8 m/s2. According to the equivalence principle, the heavier object should fall faster because its greater mass should mean a greater attractive force toward Earth but this should be balanced by the heavier object's greater inertial mass, which resists its acceleration with the result that both objects fall at the same rate. If the Unruh radiation mechanism is right, then both objects should fall a tiny bit faster than expected because inertial mass will be boosted a little bit thanks to the addition of Unruh radiation but they will fall together because the Unruh contribution depends only on acceleration and not mass. Could there be a way to select out the Unruh contribution if it exists?

The Higgs Field and Inertia

Space-time, as well as being filled with vacuum energy, is also filled with energy fields such as the electromagnetic field and the Higgs field. Interactions with the Higgs field are now thought to impart mass to particles. Could the Higgs field be involved in the mechanism of inertia as well? The Higgs field, carried out by the recently discovered Higgs boson, is thought to be a field that lends a kind of stickiness to some particles, and this is what gives them mass. It slows them down as they move through space and it prevents particles of mass from approaching light speed. Particles without mass, such as photons, are thought to be invisible to the Higgs field. They move right through it without interacting with it. At first glance, it seems that the Higgs field is a contender for inertia, until we recall that inertia is a property of both matter and energy, rather than just matter alone. The photon, having inertia but no mass, does not interact with the Higgs field.

Mach's Principle and Inertia

In the early 1900's, Einstein was not aware of the accelerating expansion of the universe when he formulated general relativity. He thought the universe was in a steady state, so it made perfect sense to try to build upon philosopher/physicist Ernst Mach's idea in order to define a universal frame of reference that is built upon on all the mass in the universe, which he thought should also be in a steady state.

This principle, created by Mach, is often described in the simplest way as "mass out there influences inertia here." A popular anecdote describes the general idea in more detail: " You are standing in a field looking at the stars. Your arms are resting freely at your side, and you see that the distant stars are not moving. Now start spinning. The stars are whirling around you and your arms are pulled away from your body. Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?" It is a loosely defined principle rather than a theory.

In the end, Einstein was not successful in fully incorporating Mach's principle into general relativity, but he was guided by it and by differing accounts, general relativity is more or less Machian in nature. Because distance and time are no longer constants, however, the question of how Mach's principle is applied to general relativity quickly becomes very complicated.

You can see some evidence of Mach's influence in general relativity in its formulation of mass as an energy-momentum metric tensor, which describes the influence of mass as both energy and momentum densities as well as stress within the four-dimensional construct of space-time. It also assigns a frame of reference that is rotationally stationary so that phenomena such as frame dragging and angular momentum can be described.

Conclusion: The Inertia Story Is Just Beginning

Inertia, a centuries-old concept, at first seems to be a long-settled dead end in theoretical physics. However, inertia is neither simple nor is it anywhere close to what we can call a complete theory. It needs a well-defined mechanism and it needs a well-defined frame of reference in order to describe it in rigorous detail.

There is no agreed-upon reference frame in which to describe inertia's action. Some formulations of inertia attempt to describe a frame of reference that relies on the original idea of Ernst Mach - that the mass of the entire universe serves as non-inertial reference frame. For example, theories such as the Mach Weber Assis theory choose an invariant frame of reference that takes into account the Newtonian gravitational constant, the Hubble constant and the mean matter density of the universe. This solution, therefore, takes into account gravitational attraction, the accelerating expansion of the universe and the density of matter in the universe (which is not a constant). Other theories, as we've seen, look to the vacuum energy of the universe as a reference frame, and as a possible mechanism.

Finally, though I have not expanded on it here, inertia poses a real puzzle in terms of locality. If a Machian view of inertia is correct, then the masses of distant galaxies affect the motion of objects here on Earth. Imagine a child on a merry-go-round. Is the mass of the entire universe trying to pull her off of it? This makes inertia a non-local phenomenon. If an Unruh radiation view is correct, then inertia originates from the local effects of a quantum vacuum on an accelerating body, and the girl is then experiencing the drag of being accelerated through a quantum vacuum.

Something seems to be missing, whether it is our current description of mass or our description of space-time. However, if these descriptions hold up, then inertia could result from an as yet unknown energy field.

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