a2290_20 (fate of the universe)

7/27/2019 A2290_20 (Fate of the Universe)

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Fate of the Universe

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Relativity and AstrophysicsLecture 20

Terry Herter

A2290-20 Fate of the Universe 2

OutlineSkateboard paradox

Fate of the UniverseCritical DensityDark Matter

Three Omegas (really two)Dark Energy

Problem L-13, page 117 (due Today/Monday)Will hand back on Monday if you hand it in today


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We have a 1-m skateboard (in rest frame of rider) and a 1-m hole (inrest frame of lab).Proper lengths are the same

A person on a skateboard moves very fast (causing relativisticcontraction). The skateboard passes over the hole. Will it fall in?Paradox

Lab person sees skateboard contracted =>should fall inRide sees hole contracted =>should pass over hole

Which person is correct?Idealize problem to a one-meter rod sliding lengthwise over a flat table.

Let the Lorentz contraction factor be 10 so that the rod is 10 cm in the labframe, thus L = L o / , where L =length of rod in lab frame, Lo is rest length of rod, and is the Lorentz factor.Assume that the skateboard does not tiltas it falls in the lab frame, that is,once it is completely over the hole it starts to drop.

Paradox of Skateboard & Grid: L-12

skateboard

Lab Frameskateboard

Skateboard Frame


Skateboard & Grid (contd)Define t =0 and t =0 when back of skateboard is even with back of hole.

In the lab frame the motion of the front and back are given by:

We now transform to the skateboard frame. For distances

Then times

Thus the front starts to fall before the back. The paradox is removedsince the skateboard drops over the edge and goes through the hole.

Lt v Lt vt v x x relrelrelF F '

Lt v x relF t v x rel B&

Lvt vt xvt t relrelF relF 2'

t vt vt v x x relrelrel B B '

hole

Lab frameskateboard /o L L

oF L x'

0' B xAs expected

orelrelF Lvvt t 2' 1 t vt xvt t rel Brel B

2' 2' 1 rel B vt t


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Skateboard paradox (graphically)

From Sartori (Understanding Relativity). Here strings, which keep the boardfrom tilting, are cut simultaneously in lab frame (left) but not in skateboard frame(right). Note: relativity does not allow the skateboard to be infinitely rigid.


Skateboard paradox (graphically) The previous graphs picture theskateboard from the viewpointof the lab where it is alwayshorizontal.If we start in the skateboardframe this will not be the case.

At left is an (incorrect) attemptto picture what is happeningfrom the skateboard viewpoint

When the first thread is cut theboard begins to droop

The board at second threaddoesnt knowit has been cut

The simultaneous cutting in thelab frame means spacelikesothe points cant communicate.

From Sartori (Understanding Relativity). Here strings, which keep the boardfrom tilting, are cut simultaneously in lab frame (left) but not in skateboard frame(right). Note: relativity does not allow the skateboard to be infinitely rigid.


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The universe is currently expanding.What will ultimately happen to it?


What will happen to the universe?Consider a planet; shoot a spacecraft from thesurface (single shot)

High Velocity

SpacecraftEscapes

Time

D i s t a n c e


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What happens (contd)

Low Velocity

Spacecraftfalls back

Time

D i s t a n c e

At the escape velocity the spacecraft just escapes.


The Evolution of the UniverseLooking at the evolution of the universe similar toescaping from a planet.

D i s t a n c e

Time

unbound

bound

marginallybound

Big Bang

Present Time


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The Critical DensityWill the universe keep expanding forever or stop expanding andcollapse?If the mass density is:

large eventual collapse (bound)small expansion forever (unbound)

The critical density is the dividing line between the bound andunbound cases.

For H o =75 km/sec/Mpc crit ~ 10 -29 g/cm 3, or about 6 H-atoms/m 3

This corresponds to 1 MW galaxy/Mpc 3

Define M as the ratio of the actual density to the critical density

densitycriticaldensityactual

Mbound

unbound

1

1

M

M


M and H oM determines the fate of the universe.Expanding forever vs. eventual collapse

H o determines the age of the universe.M = 0 age = 1/ H o : Empty Univ.

M = 1 age = (2/3) (1/ H o) : Critical Univ.

Weve already discussed H o.What is M? numerically


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The Age of the Universe (again)

D i s t a n c e

Time

marginallybound

(critical)

Present Time

Empty

1/ H o2/3(1/ H o)

bound

unbound


Measuring M Two ways to determine M are:Galaxy Counts -

Directly measure amount of mass in the universe.

Deceleration of the Universe -Measure how much faster galaxies were movingin the past.


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Galaxy Counts & Dark MatterCounting galaxies and adding up their mass yields M ~0.045.But... This misses most of the mass.Masses can be found from (gravity)

the motions of stars in galaxies,the motions of galaxies in clusters.

Gravity implies much more mass than we see in stars/galaxiesdark matter .

We cant see>90% of the mass in the universe, except bygravity!

MACHOs (Massive Compact Halo Objects)?WIMPs (Weakly Interacting Massive Particles)?

Correcting for the missing mass due to dark mattergives M ~0.3.


Deceleration of the UniverseWe need a standard candleto measure distancesand recessional velocities independently.It is difficult to find a standard candle.

Galaxies are different in the past. Type Ia Supernovae now appear to work very well.(Caused by accretion onto a white dwarf.)

A funny thing happened ... Type Ia SN results indicate that the universe isaccelerating!

Acceleration =>the expansion rate is increasingContrary to our initial bias that it should bedecelerating

Deceleration =>the expansion rate is slowing (due togravity)


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The Two Omegas

So now we only need to know M and .With zero (no) cosmological constant

Have M and k with k = 1 MWith a non-zero cosmological constant

Have M, , and k with k = 1 M

Determine Dynamics and GeometryM and set acceleration/deceleration

k sets geometry ( k = 1 M )


0

D i s t a n c e

Time

marginallybound

Present Time

empty

Expansion velocity in the past depends upon M and . Galaxies wouldbe moving faster in the past for a non-empty universe.

Accelerating

bound

unbound

Look back in time


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a2290_20 (fate of the universe)

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