Moving clocks run slow
Special Relativity
Animated version courtesy Andrew Hamilton
HW 1 due Thursday at start of class. Start reading Thorne Chapter 2
How strong is time dilation?
Depends on Lorentz factor γ =
Easy to visualize with light, but nothing special about this type of clock…
How strong is time dilation?
1
1− v2 c2
Clock moving at speed v relative to us runs slow by a factor of γ€(“gamma”) If v << c, then v2 / c2 is very small and γ ~ 1
γ=
e.g. v = 0.1 c
€
1
1− v2 c2
γ = 1 / 1− (0.1c)2 / c 2 = 1 / 1− 0.01
= 1 / 0.99 = 1.005
A clock moving at 1/10 the speed of light runs slow by about 0.5% -­‐ 18 seconds per hour
How strong is time dilation?
γ=
v € 0.5 c 0.9 c 0.99 c 0.999 c γ
1
1− v2 c2
1.15 Fastest astronomical 2.29 jets have γ ~ 100 7.09 (produced by black holes!)
22.4 Experiments
cosmic rays
Impact of Cosmic Rays with upper atmosphere creates muons (heavy electrons) !
Unstable: lifetime at rest is 2.2 µs !
!
Experiments
If they travel at almost c, distance before they decay is
L = ct
= 300,000 km/s × 2.2 × 10 −6 s
= 0.66 km
Muons should decay before reaching surface
€
Muons are produced at v ~ 0.995 c (γ ~ 8). Variation of flux (number) with height is consistent with time dilation by this factor!
Length contraction
Objects are contracted by a factor γ in the direction along which they move
e.g. the muons in the previous experiment “see” the Earth’s atmosphere as being much thinner than we measure it!
Time dilation implies distance contraction
Imagine a 1-way trip to the Sun at 0.87c
• Distance to Sun: 93 million mi. = 8 light-­‐minutes • Travel time (Earth clock): 8 min/0.87 = 9.2 min • Travel time (astronaut clock): 4.6 min due to time dilation • Distance traveled (according to astronaut): 0.87x4.6 min= 4 light-­‐minutes= 46.5 million mi.!
Distance to Sun is contracted according to the astronaut
“Lorentz” contraction is symmetric …
Astronauts see:
The Drive-­‐Thru Paradox
Imagine a relativistic ...
46,500,000 miles
… trying to visit a
v = 0.87 c
100 ft
Earth observer sees:
… without slowing down!
93,000,000 miles
v = 0.87 c
50 ft
The Drive-­‐Thru Paradox
Imagine a relativistic car ...
The Drive-­‐Thru Paradox
Imagine a relativistic car ...
… trying to visit a
… trying to visit a
… without slowing down!
…but that’s not how the driver sees it
Because it’s moving relativistically it can fit under the carport… Relativistic addition of velocities
The Drive-­‐Thru Paradox
baseball hit from moving train at velocity v
Q: How can both be true?
A: Relativity of simultaneity. train velocity V Server finds car fits under roof simultaneously. Driver thinks front end is under roof while back end hasn’t arrived yet.
What velocity do we measure for baseball?
v2 =
€
v +V
vV
1+ 2
c
As long as v and V are < c, we never see v2 faster than light
Relativistic addition of velocities
Relativistic addition of velocities
v = 0.5 c
v = 0.9 c
V = 0.5 c
V = 0.9 c
What velocity do we measure for the baseball?
v2 =
v +V
= 1 c /1.25 = 0.8 c
vV
1+ 2
c
€
v2 =
v +V
= 1.8 c /1.81 = 0.994 c
vV
1+ 2
c
€
Relativistic addition of velocities
v = c
V = c
v2 =
€
What velocity do we measure for the baseball?
v +V
= 2 c /2 = c (1+1=1!)
vV
1+ 2
c
Special relativity results (so far)
• observers in relative motion disagree about simultaneity of spatially separated events -­‐ “mixing” of time and space • moving clocks run slow • moving rulers are short • cannot add velocities to exceed c -­‐ speed of light is an ultimate speed limit
Finite speed of light + mixing of space and time create odd optical effects for v ~ c
Movies here by Ute Kraus
http://www.spacetimetravel.org/
In “reality”, geometric distortions plus:
• redshift (light from objects coming toward us shifted to blue, receding to red) • “beaming” – light from a fast moving object is concentrated to forward direction
Effects needed to understand emission from actual astronomical sources!
• gamma-­‐ray bursts • disks of gas near black holes
http://gamelab.mit.edu/games/a-­‐slower-­‐speed-­‐of-­‐light/
For fun: “A slower speed of light”
In “reality”, geometric distortions plus:
• redshift (light from objects coming toward us shifted to blue, receding to red) • “beaming” – light from a fast moving object is concentrated to forward direction
Effects needed to understand emission from actual astronomical sources!
• gamma-­‐ray bursts • disks of gas near black holes
Equivalence of mass and energy
Newton
Einstein
Mass and energy are entirely different things:
• Mass -­‐ inertial mass -­‐ gravitational mass • Energy -­‐ related to independent properties of the object (how hot it is – “thermal energy”, how fast moving – “kinetic energy” etc)
Consequence of special relativity: energy and mass are equivalent
Energy is proportional to the mass, with the scaling factor being speed of light squared:
E = mc 2
€
Not at all obvious!
Even less obvious when we work out numbers! E = 1 kg × ( 3 × 10 8 m s-1 )
2
= 9 × 1016 Joules
How big is a Joule? A power of one Watt is one Joule per second …where is the energy?
A power of one Watt is one Joule per second Total US electrical production per day is about 4 x 1016 Joules E = 1 kg × ( 3 × 10 8 m s-1 )
2
= 9 × 1016 Joules
this energy is equivalent to ~0.5 kg of matter 9000 Watts: if run for one hour – 9000 Joules per second x 3600 seconds = 3.24 x 107 Joules
Consequences
Efficiency of nuclear reactions
How long would the Sun last if it was burning gasoline?
About 10,000 years
In the Sun, nuclear fusion
4 x
protons (hydrogen nuclei)
2 protons + 2 neutrons (helium nucleus)
Mass of a helium nucleus is 0.7% less than the mass of 4 protons: reaction releases energy
0.993 kg of helium
4 x
1 kg of hydrogen
energy equivalent of 0.007 kg = 6.3 x 1014 Joules
Enormous energy: 600 million tons per second of hydrogen fusing to helium provide the Sun’s luminosity (power)