General Relativity Topics Philosophical origins A new view of gravity Gravity as a tidal force The field equations Consequences of general relativity The.

General RelativityTopics

Philosophical originsA new view of gravityGravity as a tidal forceThe field equationsConsequences of general relativityThe twin paradox truly resolvedGeneral relativity and cosmology

MotivationLearn about general relativity.

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In 1907, Einstein was asked to publish a summary of his theory of invariance (special relativity). He felt special relativity was consistent with all of physics except for gravity.

Some examples of how gravity is not consistent with special relativity…

1. If we made the Sun disappear, Newtonian gravity predicts the Earth would instantly go careening into space—this is information conveyed faster than light.

Ultimately, this is due to the mysteries of “action at a distance,” which even Newton did not like.

2. Gravity is an inverse square law force. But according to special relativity, spatial distance is a function of velocity (i.e., Lorentz contraction). How can gravity be a sensible force if it relied upon a frame-reliant parameter such as distance?

Leading to general relativity

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It took Einstein six weeks of focused work to develop special relativity. He based it on a single postulate:

Postulate of special relativityPhysical laws are the same for all observers in inertial reference frames

It took him nine years to figure out how to incorporate gravity; when he realized his key idea, “der gluecklichste Gedanke meines Lebens” (it was the happiest thought of my life):

Einstein’s happy thoughtIf a person fell off the roof of a house, they would not be able to feel the effects of gravity during the fall.


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Einstein’s gluecklichste Gedanke led him to build general relativity upon the equivalence principle, which results in a new way of thinking about gravity, and in turn, spacetime:

Equivalence PrincipleA uniformly accelerating reference frame is indistinguishable from a gravitational field.

Let us see the equivalence principle in practice, with three examples…


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Example #1: free fall is equivalent to weightlessness

Suppose you followed Einstein’s happy idea, and jumped off a roof.

For a few seconds you would plummet downwards, as you offered no resistance to gravity.

If you were holding a ball, you could let it go and it would float “weightlessly.”

Accelerating in free fall conditions ≡ zero gravity

Three examples of the equivalence principle

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A free-fall condition is not a simulation of weightlessness. It is exactly equivalent to weightlessness.

Example #2: acceleration = gravity

Suppose you were in a windowless rocket.

You measure yourself with a scale and find that your weight is normal.

There is no way for you to determine whether you are accelerating in deep space, or sitting on the surface of the Earth in 1g.

Acceleration ≡ gravity

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Acceleration does not simulate the affects of gravity. It is exactly equivalent to gravity.

Example #3: equivalent effects on a beam of light

You are in a room and a horizontal beam of light enters through a little window. Instead of following a straight line, the light bends towards the ground before it strikes the far wall. There are two ways to explain this.

A) Your room is accelerating upwards. As you accelerate upwards, the light beam bends downwards. The deflection is due to your acceleration.

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Example #3: equivalent effects on a beam of light

B) Your room is stationary with respect to the beam, but a gravitational field is bending the light. The deflection is due to gravity.

These two explanations are equivalent, and lead to a prediction of general relativity:

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Gravity bends beams of light.

In studying special relativity, we learned that if you enter a different constant-velocity reference frame, you can make an object’s velocity disappear.

Hence, velocity does not exist as something everyone can agree upon. It is a consequence of shifting reference frames.

In general relativity, if you enter an accelerating reference frame, you can similarly affect gravity—you can even make gravity’s affects disappear.

Therefore, gravity is not an element of objective reality.

Gravity can be considered to be a fictitious force.

New interpretation of gravity

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Consider a car turning on a curved path. It would love to take the easy path and go in a straight line. Forcing it to follow the curved path puts the car in a strange frame of reference. In this frame of reference, a fictitious force appears.

You call this fiction the centrifugal force.

Similarly, the structure of space is not flat, like a carefully constructed framework.

Space is warped. Objects moving in warped space follow warped paths. If you insist that objects should travel in straight lines, you will be surprised when they follow warped paths.

You must create a fictitious force to account for this strange behavior.

You call this fiction the gravitational force.


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Why does an object traveling in a gravitational field follow a curved path?

It is not following a curved path—it is moving in a straight line. The effect of gravity is to curve space.

A straight line in curved space looks curved!


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Straight lines in curved space are called geodesics—the path that takes the least amount of time to traverse from point A to point B.

Geodesics are not always easy to figure out if the curvature of space iscomplicated.

Geodesics in non-flat space

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One can use Einstein’s perspective of space to rewrite Newton’s first law:


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LAW #1 (Newton): An object at rest stays at rest, unless acted upon by an external, unbalanced force.

Similarly, an object in motion continues in motion, unless acted upon by an external, unbalanced force.

LAW #1 (Einstein): An object in warped space, allowed to fall freely, will stay at rest as viewed from the perspective within that freely falling reference frame, unless acted upon by an external, unbalanced force.

Similarly, an object in motion in warped space continues in motion in a geodesic path, unless acted upon by an external, unbalanced force.

Letting gravity have its way

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Consider the advantages of this perspective.

Imagine looking out a window, studying the trajectory of a ball shot out of a rocket.

Now imagine you are in free fall, but are (oddly enough) holding a window frame, and looking through it as you fall.

What do you observe?

Letting gravity have its way

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In this downwards-accelerating reference frame, the ball follows a straight line trajectory, as if we were in an inertial frame. Consider five snapshots taken during the ball’s trajectory.

What you might think of as a force, i.e., “gravity”…

Can be made to go away in a free-fall reference frame.

Einstein then reasoned…if gravity as we normally conceive it is fictitious, are there any effects from gravity that cannot be frame-shifted into non-existence?

Consider two balls traveling in parallel paths in space. As they approach a planet, the two balls accelerate towards the planet because of gravity.

Next, gravity pulls both towards the center of the planet, to what will eventually be a collision with each other (and the planet)…

Fine for Newton, but WRONG for Enstein. Try again, but from the perspective of general relativity!

Effects of gravity v. 2.0

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Consider two balls traveling in parallel paths in space. As they approach a planet, the two balls accelerate towards the planet because of gravity.

Next, gravity pulls both towards the center of the planet, to what will eventually be a collision with each other…

Next, their geodesic paths both lead towards the center of the planet, to what would eventually be a collision with each other…

The objects, moving at uniform speeds on geodesic paths (through warped space), merely happened to collide.


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travel at constant speeds, following geodesic paths in warped space.

What would the two balls see, in their natural free-fall frame of reference?

They would observe that they slowly drift towards each other and then collide.

Collisions are undeniable events, and are not merely a consequence of weird reference frames.

Einstein interprets this is due to travellingon geodesics in curved space. No forcesneed be invoked.

But any student of geometry will tell you that “straight lines” on a sphere will intersect.

Balls moving on a sphere would collide.


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Einstein proceeded by modeling whatever might cause the balls to come sliding together. His solution was to model the differential aspects of gravity—how strong gravity was at one place in space, compared to how strong it was at another, nearby location.

We are familiar with differential aspect of gravity, which we call tidal forces.

It was by treating tidal forces, which cannot be frame-shifted away, that Einstein developed general relativity.


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As an aside, there is an interesting historical storyline

involving Einstein and Hermann Minkowski.

In the 1890s (when Einstein was ~20), he was aninfuriating student. Minkowski, his mathematics instructor at the ETH (Zurich), called him a “lazy dog.”

In 1908, two years after Einstein published special relativity, Minkowski rewrote the theory in amathematically elegant and beautiful form.

Einstein disliked Minkowski’s formulation and he frequently made fun of it, saying it made relativity so difficult even physicists couldn’t understand it.

Four years later, Einstein ate his words—he realized Minkowski’s spacetime was an essential and fundamental component of building general relativity.

He spent many painful years formulating his physics in curved spacetime.

Minkowski and Einstein

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In their simplest form, the field equations of

general relativity can be written as:

Gμν is called the Einstein tensor, and describes the various curvature elements of space.

Tμν is the stress-energy tensor and describes the distribution of energy and matter.

Tensors are matrices, so the equation above is actually a system of equations; in this case, 10 coupled nonlinear partial differential equations.

Each separate equation is identified by the μ, ν index numbers.

(Note: often you will see systems where G and c are set to 1.)

Einstein’s basic field equations—the math

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Let us step away from the details for a moment. The key revelation of this equation is that “spacetime tells matter how to move; matter tells spacetime how to curve” (Wheeler, 1990).

Changes in the distribution of matter and energy (Tμν) change the structure of spacetime (Gμν); those changes in spacetime ripple through spacetime at the speed of light.

Matter responds to the spacetime that the matter sits in.

Matter gets its marching orders locally.


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There is no such thing as action at a distance!

Gravity is exposed as the fiction it is!

Matter and energy (Tμν) in the Universe warps spacetime.

Spacetime warpage (Gμν) forces matter and energy in the Universe to move on geodesics.

In the next level of complexity, the Einstein tensor can be expanded:

Rμν is the Ricci curvature tensorIt describes how spacetime is different from Euclidean spacetime.

gμν is the metric tensorThis is the key feature you are studying in general relativity. It determines the nature of space, and defines quantities such as distance, volume, curvature, angles, and how rapidly time flows.

R is the scalar curvature

This is a spatially varying value, similar in information to Rμν.

Λ is the cosmological constantThis is an arbitrary value that can be added to the equations.


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Einstein’s equations are so complex, they can be solved only for certain highly simplified cases.

For example, the first solution of Einstein’s equations were made by Karl Schwarzschild for a perfectly symmetric, non-rotating star with a uniform interior.

His second solution was for the star’s interior; only much later was it realized that this solution was for a non-rotating black hole.

Even the case of gravity around an axially symmetric object becomes horribly complex.

In the current era, numerical methods are used to solve equations that could never have been solved in the past.

I vote we avoid the math, and keep the discussion general—OK?


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1) Light escaping from a gravitational field becomes redshifted!Imagine you release a mass (m) from a tower on Earth. Starting at zero speed, the mass’s initial energy is E = mc2.

It falls a distance (h) in the gravitational field to a receiver on the ground. At the bottom, its energy is E = mc2 + mgh.

The receiver converts the mass into a photon with the same total energy: E(γ) = mc2 + mgh.

The photon is fired upwards, back to the top of the tower.

The receiver at the top converts the photon back into matter. The blob’s new mass is m = E(γ)/c2 = m + mgh/c2.

This is more massive than it was before!

The only way around this gain of mass is if the photon loses energy on the flight upwards, out of the gravitational well!

Four new implications of general relativity

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m

2) Time runs slower in a gravitational field!

It is easy to imagine that the structure of space is warped by the stress-energy tensor.

But, as Minkowski demonstrated, space and time are intermixed. As space is warped, so is time.

10,000 km above the Earth, clocks run slightly faster than those at the Earth’s surface, by the difference of 4.5 parts per 1010.

→ Earth clocks run slow by 1 sec every 70 years.

In 1976, the Smithsonian Astrophysical Observatory shot a Scout rocket to 10,000 km; during two hours of free fall it transmitted pulsed signals to the ground, the lengths of the time intervals (after corrected for special relativity) confirmed general relativity to within 0.01%.


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2) Time runs slower in a gravitational field!

Global Positioning Systems must correct for relativity!

GPS depends upon receiving signals from satellites; the timing of the signals must be known to within ~25 nanoseconds.

But satellites are moving rapidly (14,000 km/hr) and are far from the Earth (about 27,000 km from the Earth’s center).

In a single day, compared to a person on the ground, special relativity says that a satellite clock (moving rapidly) would lose 7 microseconds; general relativity says a satellite clock (in comparatively low spacetime warpage) would gain 45 microseconds.

The difference (38 microseconds/day) means a GPS fix would be accurate to only a few kilometers if it did not correct for both special and general relativity.


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3) Gravity bends the path of light

For neophytes who do not know that light travels on geodesics, this is a surprise. We know better, and regularly use gravitational lenses as a tool in astronomy.

4) Rotating masses drag spacetime with them

This effect is called frame-dragging. This means an object falling straight towards a rotating mass will start moving sideways before it reaches the surface. The resulting lateral fictitious force is called gravitomagnetism or gravitoelectromagnetism. It has been verified by a space mission called Gravity Probe B.


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Remember the Twin Paradox?

In summary:Rocket-Ronda and Spacestation-Sally are both 20 years old. Rhonda flies at 0.8c to a point 8 LY away and then returns.

Spacestation-Sally’s PerspectiveRocket-Rhonda took 10 years (t = 0.8 LY/0.8c) for each leg of the trip. Spacestation-Sally is therefore 40 years old when Rocket-Rhonda returns.

Because of time dilation, Rocket-Rhonda will age only 20/γ years (12 y) during her high velocity trip. When the two twins reunite, one’s age is 40, and the other’s is 32.

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Remember the Twin Paradox?Rocket-Rhonda’s PerspectiveTraveling at 0.8c, the distance to the turn-around point is Lorentz-contracted to 4.8 LY (i.e., 8/γ LY). It takes 6 years (4.8 LY/0.8c) for each leg of her trip, so she is 32 years old (20+6+6) upon her return (this agrees with Spacestation-Sally’s perspective).

But….in Rocket-Rhonda’s frame, Spacestation-Sally was the one who was moving.

During Rocket-Rhonda 12-year trip, she observes Spacestation-Sally’s to age only 7.2 years (12/γ years). Upon their reunion, 32-year-old Rocket-Rhonda expects to find her sister to be only 20+7.2 = 27.2 years old. Imagine her surprise, when she meets a woman who is 40!

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In our study of the paradox, I pointed out that “Rocket-Rhonda accelerates at the turn-around point of her travel. Accelerating reference frames are not treated by special relativity.”

If Rocket-Rhonda filmed Spacestation-Sally for the entire trip, what would she see?

1) During the 6-yr outward bound trip, Spacestation-Sally ages only 3.6 yrs (6 yr/γ).

2) At the turn-around point, Rocket-Rhonda decelerates hard, then she turns around and accelerates hard, again. Although this takes her but a few instants, her accelerating reference frame is equivalent to being in a huge gravitational field.

Spacestation-Sally (and the rest of the Universe) ages by 12.8 yrs in the tiny amount of time it takes for Rocket-Rhonda to reverse direction!

3) During the 6-yr return trip, Spacestation-Sally ages only 3.6 yrs. When the two reunite, Spacestation-Sally is 40, and Rocket-Rhonda is 32.

And that is the real resolution to the Twin Paradox!31

Remember the Twin Paradox?

Mercury’s orbit

Mercury’s perihelion precesses over time. After all known sources of gravitational perturbations are removed, there is a tiny, residual: 0.431”/year, taking 3 million years for the orbit to precess once.

After a number of attempts, Einstein was able to explain this with general relativity in 1915.

Deflection of light

In 1915, Einstein predicted that starlight would be deflected as it passed by the Sun. In 1919, during a total solar eclipse of the sun, Sir Arthur Eddington was able to photograph the Sun and nearby stars. He observed a deflection of about 1.7”.

General relativity is tested on a daily basis; gravitational time dilation, redshifts, lensing are all readily measured.

Proofs of general relativity

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The Universe is highly uniform, with a simple stress-energy tensor permeating all of space. The field equations have been solved for the Universe, with different simplifying approximations.

A few examples…

Friedmann–Lemaître–Robertson–Walker (FLRW) solutionsThe standard model, developed in 1920-30. This can lead to the Friedmann equation:

G = gravitational constantρ = density of matter and energyH = Hubble constantK = curvature of space

De SitterA Universe empty of matter, and only driven to expand by dark energy (or its equivalent). Our Universe is heading in this direction.

General relativity and cosmology

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The amount of energy and matter of the Universe, compared to how fast it is expanding, will determine if its density exceeds the critical density ρcrit.

The value of ρcrit = 10-29 g/cm3, or about 1 H atom/m3.

Positive curvature (K > 1), ρ > ρcrit

– The Universe is finite;

– The Universe will collapse upon itself.

Negative curvature (K < 1), ρ < ρcrit

– The Universe is infinite;

– The Universe will expand forever.

Zero curvature (K = 1), ρ = ρcrit

– The Universe is infinite;

– The Universe will expand forever, but just barely!

The curvature of the Universe

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Positive curvature, ρ > ρcrit

– The sum of interior angles in a triangle >180º;

– Parallel lines cross.

Negative curvature, ρ < ρcrit

– The sum of interior angles in a triangle <180º;

– Parallel lines diverge.

Zero curvature, ρ = ρcrit

– The sum of interior angles in a triangle =180º;

– Parallel lines stay parallel.

The curvature of the Universe

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We can measure the spatial curvature of the Universe directly by using a standard ruler. A standard ruler is an object of a calibrated size.

Standard rulers are analogous to standard candles, such as Type Ia supernovae.

Standard rulers are hard to come by, since everything large in the Universe tends to come in a range of physical sizes.

Furthermore, since the curvature of the Universe is subtle (at most), you need a very large shape at a very large distance to detect the tiny amounts of curvature.

Measuring the curvature of the Universe

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Suppose an object of size (L) is a distance (D) away: its apparent angular size (θ) would depend upon the curvature of space.

If the Universe is flat, the geometry is simple: tanθ = L/D.

But if space is positively curved, tanθ > L/D.

If space is negatively curved, tanθ < L/D.


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LD

θ

θ

θ

Consider collapsing lumps of gas in the early Big Bang (i.e., before the recoupling, t=380,000 years).

Very large lumps did not have time to collapse, meanwhile small lumps would have collapsed, then rebounded as they heated.

Lumps of size L = ct = 380,000 LY would show the maximum amount of density enhancements. These would appear as the most common, most intense variations in the cosmic microwave background.

This is a very good standard ruler, and at this distance, such lumps should be about 1º (flat), >1º (positive curvature), <1º (negative curvature).

The lumps in the cosmic microwave background are 1º.

The overall warpage of the Universe’s spacetime is flat.


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Those who are familiar with calculus know that when integrating equations, you develop a “constant of integration.”

The value of this constant must be determined by comparing your result to measurements.

When Einstein solved his equations for simple cases, his equations predicted the Universe was expanding. He set a cosmological constant to counter this expansion.

Cosmological redshifts were discovered by Hubble (1929), and Einstein concluded that Λ=0.

We shall see, with dark energy, that the cosmological constant was revived.

The cosmological constant

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General relativity predicts some very weird things…

Black holes

Gravitational radiation

Extreme relativistic effects with binary pulsars

Wormholes and warp bubbles

More about these at the end of the semester!

Extreme general relativity

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General Relativity Topics Philosophical origins A new view of gravity Gravity as a tidal force The field equations Consequences of general relativity The.

Documents

acceleration gravity

effects of gravity

new view of gravity

affects of gravity

newtonian gravity

equivalent effects

light bends

einsteins happy thought