Gravity and Energy

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Figure 1: Mass-Energy Venn diagram

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Gravity and Energy

Exploring the interplay of gravity and mass-energy

P. Lutus Message Page

Copyright 2010, P. Lutus

Introduction | Conservation of Mass-Energy | Escape Velocity

Cosmological Implications | Notes

(double-click any word to see its definition)

Introduction

n this article we'll explore the relationship between gravityand energy, and consider some consequences for matters

both large and small. This article uses animations and graphicsto clarify its points, and some key equations are included andexplained. Finally, we'll discuss a new theory about the universe how it might have come into being without either violating anylaws of physics or requiring supernatural intervention.

The overview:

In modern physics, mass and energy are complementaryaspects of a fundamental quantity that, for lack of a betterword, we call mass-energy.Mass-energy cannot be created or destroyed, onlychanged in form. This is called the Principle of EnergyConservation (some use the term "law").Energy has two basic forms kinetic andpotential .Kinetic energy is the energy of motion examples mightbe a spinning wheel or an arrow in flight.Potential energy is the energy of position or state examples might be a book on a high shelf or a chargedbattery.Many physical processes cause energy to be convertedfrom potential to kinetic or the reverse, and from energy tomass or the reverse.

The unit of power is the Watt . One watt may be definedin several ways. Here are two:

A constant velocity of one meter per second against an opposing force of one Newton .A current flow of one ampere through a potential difference of one volt.

The energy unit is the Joule .Energy is the time integral of power. One joule is defined as the expenditure of one watt of power for one second.Mass and energy are complementary aspects ofmass-energy:

To convert mass to energy, use this equation:

(1)

To convert energy to mass, use this equation:

(2)

Mass has units of kilograms .Energy has units of joules .The constant c in the above equations is the speed of light and is equal to 299,792,458 m/s.

These principles are not merely laboratory curiosities, they're part of everyday life:If I lift a one-kilogram book from the floor and place it on a two-meter-high shelf, the book gains 19.6 joules of

potential energy (enough to power a small flashlight for about one second) and 2.2 * 10-16 kilograms of mass(about 1/3 that of a small bacterium).If I take a fully discharged D-size flashlight battery and charge it fully, it gains 74,970 joules of potential energy and

8.3 * 10-13 kilograms of mass, about that of a typical human cell.Kinetic energy is relatively easy to quantify with physical measurements. It is equal to:

(3)

ek = kinetic energy, joulesm = mass, kilogramsv = velocity, m/s

Potential energy has much more variety and is a bit more difficult to pin down. One of its simpler forms comes up in agravitational field, where it is equal to:

(4)

ep = potential energy, joules

G = universal gravitational constant , equal to 6.67428 * 10-11 m3 kg-1 s-2

m1,m2 = masses (kilograms) of two bodies in mutual gravitational attraction.

vity and Energy http://www.arachnoid.com/gravity/index.html

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(Click below to start or stop the animation)

Figure 2: pendulum energy model

r = distance between m1 and m2, meters.

Power and Energy

Notice the minus sign in equation (4) above it means that gravitational potential energy is negative. Because this is animportant property with cosmological significance, I would like to explain how it comes about.

The reader may recall my earlier remark than energy is the time integral of power, but this is just one example inmechanics, work (energy) can be expressed as the integral of force with respect to distance (x) rather than time:

(5)

Expressed in everyday language, work is equal to force times distance. Now we'l l apply this to gravitation here is the forceequation for gravitational attraction between two masses m1 and m2, separated by a distance r, and under the influence of thegravitational constant term G:

(6)

Equation (6) is the classic expression of Newton's Law of Universal Gravitation . To move from force to energy, we need tointegrate equation (6) with respect to distance (r):

(7)

Equation (7) tell us that negative gravitational potential energy is the correct physical interpretation, and it arises frommathematics, not an arbitrary choice or convention.

One more thing under General Relativity , gravity is not a force, instead it arises as a result of spacetime curvature. But inordinary circumstances the Newtonian conventions still apply, and energy is still a meaningful concept in orbital mechanics.

Conservation of Mass-Energy

NOTE: If the animations in this section distract the reader, one may click them to make them stop.

emember that mass-energy cannot be created or destroyed, only changed in form. A more general way to say this is thatthe universe has a constant quantity Q of mass-energy, fixed at the moment of the Big Bang and unchanged since. We'll

be discussing the quantity Q throughout this paper, and we'll eventually assign it a value.

Pendulum

As a mass moves in a gravitational field, it typically exchangeskinetic and potential energy. A swinging pendulum (Figure 2)has maximum kinetic energy at the lowest point in its swing,and zero kinetic energy at the highest. The pendulum'spotential energy has the reverse relationship it increases(i.e. becomes less negative) with distance from the center ofthe earth, and in exchange, the kinetic energy must decrease.The important thing to understand about freely movingobjects in a gravitational field is that their energy, the sum ofkinetic and potential energy, is constant.

There is a well-known principle in mechanics calledNewton's First Law which says that, unless acted on by anexternal force, an object will maintain a constant state ofmotion. There is, or should be, a corollary for freely movingobjects in space:

Unless acted on by an external force, an object movingin space will maintain a constant energy.

This doesn't mean the object's velocity will remain the same,nor does it mean the object's kinetic and potential energyvalues will remain the same. It means the total energy, thesum of kinetic and potential energy, will remain constant.

The swinging pendulum in Figure 2 shows this even thoughthere is a periodic exchange between kinetic and potentialenergy, the total energy (ek + ep) is constant. If our

pendulum were located in a vacuum and had losslessbearings, it would continue to swing forever in the same way, perpetually conserving its energy. (In Figure 2, the height of thered/green bar at the left represents the sum ofkinetic and potential energy. Because the pendulum's energy is constant, thebar's red and green sections always sum to the same height.)

For small-scale mechanical systems like the pendulum, it's convenient to establish an arbitrary zero point for potential energy.In this case, the zero point is set at the bottom of the swing, so potential energy is pictured as increasing from zero to positivevalues as the pendulum swings. This is a reasonable way to picture a physical system, but the absolute value of gravitationalpotential energy is typically a much larger value, and is always negative.

Satellite

Pendulums don't usually get to swing in a vacuum with frictionless bearings, but an orbiting satellite is a better example of africtionless system. Like the pendulum, as the satellite orbits it carries both kinetic and potential energy:


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(Click below to start or stop the animation)

Figure 3: elliptical orbit energy model

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Its (positive) kinetic energy results from its orbital velocity.Its (negative) potential energy results from its altitude above the center of mass of the body it orbits.

Here again are the equations for kinetic and potential energy (ek and ep), and a derived equation for total orbital energy (et):

Kinetic energy ek Potential Energy ep Total Orbital Energy et

(8) (9) (10)

To be consistent with the Principle of Energy Conservation , for a freely orbiting body with no external forces acting on it,equation (10), the sum of kinetic and potential energies, always produces a constant.

Figure 3 shows a satellite in a very elliptical (oval-shaped) orbit around a central body. I chose this configuration to show that,even though there is an ongoing exchange between kineticand potential energy, as with the pendulum the total energyremains constant. By the way, this orbital shape isn'thypothetical comets often have elongated orbits l ike this.Many comets dwell far beyond Pluto and only rarely descendinto our neighborhood for a brief appearance.

Interestingly, Johannes Kepler computed the properties oforbits and wrote what we know as Kepler's Laws ofPlanetary Motion , but without understanding the reasonorbits behave as they do. The secret to understanding orbitsis to recognize that their motion conserves energy, and if theybehaved at all differently, nature would need different laws.

Now let's look at the relationship between an orbit's kineticand potential energy. Here is an approximate equation for thevelocity of a circular orbit vo:

(11)

Where m1 is the satellite's mass, m2 is the central body'smass, r is the orbital radius and G is the universalgravitational constant. If the satellite is much less massivethan the central body, this approximate equation is suitable:

(12)

Where M is the central mass, r is the orbital radius and G is the universal gravitational constant. It turns out that, for circularorbits, the relationship between kinetic and potential energy is fixed, regardless of the orbit's other properties the negativegravitational potential energy is always twice the magnitude of the positive kinetic energy. Another way to say this is that, for acircular orbit, 2/3 of the energy is negative potential and 1/3 is positive kinetic.

Escape Velocity

n each of the cases examined so far the pendulum as well as the elliptical and circular orbits the sum of energies hasbeen negative, dominated by negative gravitational potential energy. Obviously we might select a very high velocity and

produce a positive result for equation (10) above, the total orbital energy. But is there an orbital velocity that exactly balancesthe two kinds of energy and produces zero? Yes, there is it's called escape velocity . Here is its value, using the terminologyfrom the previous section:

(13)

Escape velocity has some interesting properties. If an object is propelled away from an airless planet with an initial impulse ofescape velocity (sort of like Alan Shepard's famous golf shot on the moon, but a much higher velocity), that object willcontinue to move away, at gradually decreasing speed, but it will never stop and return. In fact, at an infinite distance, anescape-velocity object will achieve zero velocity. Here are the properties of an object initially given escape velocity:

The object has zero energy positive kinetic energy ek and negative gravitational potential energy epare equal.At an infinite distance, the object will achieve zero velocity.Therefore escape velocity is the only case where, at an infinite time and distance, an object possesses zero energy andachieves zero velocity.

There are two canonical orbital velocities one is the circular velocity vo provided by equations (11) or (12), the other is

escape velocity (ve) provided by equation (13). Velocities greater than escape velocity ve or less than circular velocity voproduce interesting effects, like the large family of elliptical orbits resulting from initial velocities in the range 0 < v < vo and

shown in Figure 3 above. But because escape velocity has properties of cosmological significance, it merits a closer look.

Because of its importance to what follows, we should prove that escape velocity results in zero net energy (i.e. ek + ep = 0).First, let's simplify equation (10) let's normalize the mass of the orbiting body m1 to 1. Here is the result:


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W

(14)

Remember that equation (14) provides the total energy of an orbiting body, the sum of positive kinetic and negative potentialenergy. At this point, those sufficiently adept at mathematics will compare equations (13) (escape velocity) and (14) (totalenergy) and a light bulb will go off. For the rest of us, here's a step-by-step proof:

(15)

Q.E.D. An object given escape velocity will have zero orbital energy, and very important, this is only true at escape velocity, noother.

Cosmological Implications

Big Bang

hen Georges Lematre first proposed the Big Bang theory, there were a number of objections at the time therewas no evidence in support of the idea, it seemed counterintuitive, and it appeared to violate basic physical principles.

How could the universe arise out of nothing?

But over the years, evidence has begun to accumulate that the Big Bang may be rea l:

Astronomer Edwin Hubble detected a systematic redshift in the spectra of distant galaxies (more distant galaxies have

proportionally more redshift), and this was eventually taken to mean those galaxies were moving away from each otherand us.Physicist George Gamow conjectured that, if the Big Bang was real, there would be a residual radiation left over fromthe exceedingly high temperatures of the explosion, but very much redshifted, all the way into the microwave region ofthe spectrum and with a characteristic temperature of about 5 Kelvins.Radio astronomers Arno Penzias and Robert Wilson inadvertently discovered this microwave signal, now known as thecosmic microwave background (CMB) radiation , and this signal has become the subject of intense study.

All this evidence has given the Big Bang the status of a scientific theory, that is to say, a theory supported by evidence andfalsifiable in principle. But one objection to the Big Bang remains, and it is serious there is no physical law sowell-established as the conservation of mass-energy, and the Big Bang seems to violate it. By creating an entire universe ofmass-energy out of nothing, the Big Bang seems to break the most basic rule of physics: no free lunch.

But this final objection is answered by the idea expressed in this article if the universe began with an exact balance betweenpositive mass-energy and negative gravitational potential energy, the law of mass-energy conservation is honored.

For this condition to be met, the Big Bang would have to create the universe with an exact balance between positive

mass-energy and negative gravitational potential energy, so the total mass-energy is equal to zero. Therefore the Big Bangwould have to give matter an initial velocity exactly equal to escape velocity. Is there any evidence for this? In a word, yes.

It turns out there is a relationship between the average velocity of matter in the expanding universe, and the overallcurvature of spacetime . Since the beginning of Big Bang cosmology, the spacetime curvature issue has been much studied,with three likely outcomes:

Terms:

= Mass-energy density parameterQ = Sum of positive kinetic energy and negative gravitational energy.V = Expansion velocity with respect to escape velocity ve. = Sum of a triangle's inner angles.

Density Parameter Energy total Q

Velocity VTriangle Sum

Description Graphic (click images for 3D)

> 1.0Q < 0V < ve

> 180

Expansion velocity is lessthan escape velocity,negative gravitationalenergy predominates,space is positively curved,expansion will reverse andthe universe will eventuallycollapse.


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= 1.0Q = 0V = ve

= 180

Expansion velocity is equalto escape velocity, totalenergy is equal to zero,space is flat or classicallyCartesian, expansionvelocity will decreaseasymptotically and reachzero at infinity.

< 1.0Q > 0V > ve

< 180

Expansion velocity isgreater than escapevelocity, positivemass-energy predominates,space is negatively curved,expansion will not convergewith zero.

I emphasize the above table summarizes conditions near the time of the Big Bang. The recent discovery of Dark Energy as anacceleration term in universal expansion doesn't change the physics for that era because positive mass-energy and negativegravitational energy were both much larger factors than dark energy.

The above table suggests that, if space is classically flat or Cartesian, this supports the zero-energy condition required for theBig Bang to create the universe without violating energy conservation. And there is good evidence that space is flat. Thisdoesn't mean there isn't local strong curvature near masses, it means the overall large-scale curvature of spacetime is flat.

Quantum Uncertainty

It has been recently suggested that, if the Big Bang could impart escape velocity to the universe's matter thus balancing

positive and negative energy a random quantum fluctuation could have brought the universe into existence. To thoseunfamiliar with quantum ideas this may seem absurd aren't quantum effects limited to extremely small scales?

Well, no quantum effects are a matter of probability, not possibility. On a microscopic scale, quantum effects are routine andmust be taken into account. But there's no "quantum barrier" that separates large-scale reality from the microscopic scale. It isa simple matter of statistics the probability of a macroscopic quantum effect is inversely proportional to the mass underconsideration. Consider this expression:

(16)

Where:

x = uncertainly in position p = uncertainly in momentum

= Planck's Constant adjusted

The above relation, known as Heisenberg's Uncertainly Principle , describes the role of uncertainly in quantum theory. Insteadof denying the possibility of large-scale quantum effects, this principle gives them a probability estimate. And the outcome isthat, for large masses, one might have to wait a very long time to see a manifestation of quantum uncertainly at a macroscopicscale maybe even a billion years. But a billion years seems like reasonable time to wait for a universe.

"Because there is a law such as gravity, the universe can and will create itself from nothing ... Spontaneous creation is thereason there is something rather than nothing, why the universe exists, why we exist." Stephen Hawking in "The GrandDesign" .

Notes

Further Reading:

Conservation of EnergyNewton (unit of force)Watt (unit of power)

Joule (unit of energy)Newton's Law of Universal GravitationGravitational potential energyKepler's Laws of Planetary MotionOrbital speedEscape velocityShape of the Universe


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Dark EnergyHeisenberg's Uncertainty Principle

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