Black Holes

Any object that has an escape velocity greater than the speed of light is called a black hole.

Albert Einstein

Einstein and Black Holes

The existence of black holes is predicted by Einstein’s General Theory of Relativity.

But black holes are so strange that Einstein refused to accept that nature would allow such a thing to exist.

Was Einstein wrong?

Today most theorists believe that black holes must exist, though all the observational evidence is still indirect.

Albert Einstein

Special Theory of Relativity – 1905

Applies to objects moving at a constant velocity

Discussion

Which way is up? How would you define the concept of “up?”

Discussion

Suppose a supersonic airplane is flying at a speed of 1,650 km/hr from Nairobi, Kenya, to Quito, Ecuador. How fast is the plane going?

Velocity is relative

In order to talk about the speed of an object, you need to have a reference frame in mind. The speed something is moving must always be measured relative to something else.

Discussion

Suppose you are running on a treadmill at 8 miles/hr. How fast are you moving relative to the ground? What does it mean to say you are running at 8 miles/hr?

Discussion

I can throw a ball at 30 m/s. If I’m riding on a train traveling at 10 m/s toward you and I throw a ball to you standing by the side of the track as the train comes toward you, how fast is the ball going when you try and catch it?

Principle of relativity

Newton’s laws are exactly the same whether one is moving at a constant velocity in an airplane or at rest of the ground.

Electric and Magnetic Waves

The speed of light is 3.00 × 108 m/sec in a vacuum

Water Waves

Michelson-Morley experiment

Einstein’s postulates

1. The laws of nature are the same for everyone.

2. The speed of light in a vacuum is constant for all observers.

Discussion

If Jackie is in a space ship traveling at 1×108 m/s and she turns on her headlights, the light leaves the ship at 3×108 m/s. If you are in front of the ship, how fast would you measure the light to be traveling past you?

The speed of light

All the strange results of Relativity come from the observation that the speed of light is constant for all observers.

Consider a space ship traveling at a very high speed (maybe even faster than the speed of light). It has a light on the front of it. How fast does the light leave the ship as measured by those traveling with the ship?

Discussion

Discussion

Consider an observer watching the ship pass at a very high speed. How fast will they measure the light to be traveling that the ship has emitted?

From the perspective of those on the ship, the light must always proceed the ship. Is this true for observers watching the ship pass? What must they conclude about the speed of the ship relative to the speed of light?

Discussion

No matter can ever travel at or faster than the speed of light.

Bottom line

If I’m traveling at high speed, relative to you, with a laser, I will measure the speed of the light emitted by it to be c. If you measured the speed of that same light at the same time, you would also get c for the speed. How can this be possible?

Hint: How do we define speed?

Discussion

Speed is a distance traveled in a period of time. In order to agree on the speed of light, we must disagree on the distance traveled, or the time it took or both.

Speed of light is not relative, space and time are relative

Length and time are relative

Throwing a ball up

If you throw a ball straight up it will come straight back down.

If you are standing in an airplane in smooth and level flight and you throw a ball straight up what happens?

Throwing a ball up

If I walk at a steady velocity and throw the ball straight it how will you see the motion of the ball?

Discussion

Repeat the experiment in the boxcar with light. Remembering that the speed of light must be the same for all observers, for which observer will the light travel time be the greatest? Explain.

Light clocks

We could make a clock consisting of two mirrors a fixed distance apart facing each other. One of the mirrors is only partially silvered and allows 1% of the light to pass through. A laser can then be bounced back and forth between the two mirrors. Each time the light pulse is detected behind the partially silvered mirror, the clock records another tick.

Time dilation

From your point of view, time runs slower in the reference frame moving relative to you.

The faster it is moving the slower you observe the time to pass in the other frame.

Consider Jackie’s view of your light experiment as she speeds by you. How will she measure the time interval for the light to complete one round trip as compared to your measurement?

Discussion

Discussion

You and Jackie are each on separate trains that will pass each other at high speed. Both of you decide to measure how fast the trains pass each other. How can you do this without leaving your seat?

Discussion

Now that you and Jackie have the relative speed of the trains, both of you decide to measure the length of the other’s train. How would you do this again without leaving your seat?

How can you measure the length of your train without leaving your seat?

Discussion

Discussion

Will the length of your train that you measure be the same as the length of your train that Jackie measures? Explain.

Discussion

How will your measurement of Jackie’s train compare with her own measurement of her train? Explain.

Length contraction

From your point of view, length of an object moving by you (or distance between objects moving by you) is shorter in the direction of motion than it would be if it were at rest.

The faster the objects are moving, the shorter the lengths.

Length and time measurements depend on the relative velocity of the observers in such a way as to make the speed of light constant for all observers in any reference frame.

Theory of Invariantness?

Discussion

If you see the light from two different lightning strikes at the same time, is this good enough to prove that the strikes occurred simultaneously? Why or why not?

Simultaneous from where?

If you are a distance d from the first strike and a distance 10 × d from the second strike. The light from the second strike will take 10 times as long to reach you as the light from the first strike. Thus, if you see the light from the strikes at the same time they are not necessarily simultaneous.

Discussion

What else do you need to know to determine the simultaneity of two events not moving relative to you?

Discussion

What else do you need to know to determine the simultaneity of two events not moving relative to you?

You need to know the time the two events took place and the distances between you and the events.

A simplification

To avoid doing any unnecessary calculation, consider the situation where you just happened to be exactly at the midpoint between the two trees when the lightning strikes. In this case, seeing the light reach you at precisely the same time, requires that the two events took place simultaneously.

Relativity of simultaneity

Space fleet maneuvers

Consider three space ships moving in formation. The flag ship is midway between the lead ship and the rear ship.

The “go” order

The flag ship now signals to the other two ships to fire their engines for exactly one minute. The signal travels at the speed of light to each ship which obeys the command. The flag ship simply adjusts its own speed to remain at the midpoint between the two ships.

Because both ships are the same distance from the flag ship, both ships fire their engines at the same time. Because they are all moving together at the same speed, they all stop firing their engines at the same time and the formation is preserved.

Eat lunch

The flag ship then signals to the other two ships that it is time to eat lunch. From the moving ships perspective they are all at rest with the flagship at the midpoint. Thus both ships received the “lunch” command at the same time and lunch takes place on both ships simultaneously.

The relativity of time

For someone at rest with respect to the moving ships, the signals also travel at the speed of light, but during the time it takes to reach the back ship it moves forward, toward the signal and thus receives the command a little early. Also during the travel time for the light the lead ship has moved forward, away from the light and receives the signal a little late.

Not simultaneous

Thus, from someone at rest with respect to the moving ships, the last ship eats lunch first and the lead ship eats lunch last.

Discussion

After lunch, the flagship again gives an order for another engine burn. From the perspective on someone at rest which ship receives the order first? What happens to the formation of the ships?

Desynchronized motion

Again the rear ship receives the order first and fires its engine before the other two. Thus, the rear ship for a time is going faster than the other two ships and gets closer to the flag ship. The lead ship gets the message last and starts its engine burn later, allowing the other two ships to catch up.

The formation length has shrunk

After the lead rocket shuts down its engine, all ships will again be going the same velocity, but the distance between them is now smaller than it was before.

Desynchronized time causes length contraction

Discussion

How will this appear to the captains of the ships flying in formation? Will they see their ships as being closer? Why or why not?

Are the captains blind?

The ships can consider themselves at rest before the second engine burn. Therefore, from their perspective all the ships fired their engines at the same time and the distance between them did not decrease.

Discussion

How could the ships get closer to each other without the captains knowing it? Consider doing another engine burn. The ships will get closer still. Eventually, even if the fool captains cannot see the difference won’t they feel it as the run into each other?

Length contraction

No. The ships will never run into each other. From the perspective of someone at rest the distance between the ships will decrease to 0 as the speed approaches the speed of light. But the lengths of the ships also decrease in size so that they never hit.

Space is contracting

It is not just objects that contract, but space itself the is contracting. The spaceships contract because the space between their atoms is contracted.

Suppose you have two identical space ships, one is at rest with respect to you, the other is moving at high velocity. As the ship passes you, you give each ship an identical push for the same duration of time. How do passengers on the moving ship perceive the duration of push you give them as compared to the duration of the push you gave the stationary ship?

Discussion

Discussion

If the moving ship observers you giving the stationary ship a push for a longer duration, how does this change the acceleration they feel as compared to the acceleration they observe you giving the stationary ship? Explain.

According to Newton’s second law F = ma. If the force F applied is the same in both cases and the acceleration is less in the case of the moving ship, what does this mean for the mass of the moving ship? Explain.

Discussion

Mass increase

From your point of view, objects moving by you have a greater mass than they have while at rest with respect to you.

The bigger the velocity, the greater the effective mass

A fist fight

The force of a punch depends on how fast you can throw a punch (the speed) and the mass of your fist.

A high speed fight

Suppose two twin brothers get into a fight. Each is traveling on a train going in opposite directions with a constant velocity and each tries to punch the other as the trains pass.

Time dilation again

But it takes time to throw a punch. Each twin will consider himself at rest and he will see the other twin as moving in slow motion.

Each will also conclude that the other’s slow motion punch will have little effect on them, and that their fast punch is going to knock the other’s fist back in his mouth.

Who is right?

Which twin wins the fight?

Discussion

Neither. The punches will hit with equal force and neither is able to shove the other’s fist back into his brother’s mouth. How can the other twin’s slow motion punch have as much force as their own fast punch?

The Universal Speed Record

You’ve just built the most powerful rocket that ever existed and you’re going for the speed record. After running your rocket engines for several months you start to approach the speed of light. Because you’ve taking astronomy you know you cannot brake the speed of light but you cannot quit now. What happens if you keep your engines firing?

As the speed of and object approaches the speed of light as viewed from a bystander, time slows to a stop, its length in the direction of motion approaches 0 and its mass tends toward infinity, thereby resisting any further increase in speed.

Kinetic energy

E = 1/2mv2

At low speeds changes in kinetic energy mostly goes into increasing the speed, with little increase in mass. As the speed approaches the speed of light, little goes into increasing the speed and instead goes into increasing the mass.

E=mc2

Actually, E2 = m2c4 + p2c2 where p = mv

At rest v = 0 and we have E2 = m2c4 or:

2 4 2E = m c mc

The minus sign being used to predict the existence of antimatter.

Special Theory of Relativity

No matter can travel at or faster than the speed of light in a vacuum

Length contraction

Time dilation

Mass increase

Discussion

Given the results of the Special Theory of Relativity, why did Einstein conclude that Newton’s law of gravity needed to be revised?

Spacetime

The constancy of the speed of light, connects the three dimensions of space with time. The speed of light (measured in m/s) can be considered a conversion factor between space and time.

All observers will agree on distances measured in the 4-dimensions of spacetime.

Spacetime is not relative

4-dimensions

We live in a 4-dimensional universe and all the objects we see have 4-dimensions.

Imagining the fourth dimension

Discussion

What would a 3-dimensional sphere look like as it passed through a 2-dimensional plane?

Discussion

What would a 4-dimensional sphere look like as it passed through our 3-dimensional space?

Discussion

What does a person look like in 4-dimensional spacetime?

But, time is different, or is it?

Consider what a universe with 4 equal space dimensions and no time dimension would be like if you and me and everything else were moving very close to the speed of light in some direction.

Moving the speed of light

Lengths in the direction of motion would shrink to 0.

We wouldn’t be able to move in this fourth dimension.

We would be forever stuck at a single point.