When Albert Einstein applied his equations of General Relativity to the observable universe, he found something that he didn't like. His equations were correct, but for some reason, his equations predicted that the universe was dynamic when he and everyone else though the universe was static. This was in 1917, before the Big Bang theory and before Edwin Hubble found that the universe was expanding. To account for what he felt was incorrect, he introduced a fudge factor to take away the dynamic universe solution. He called his fudge factor, the cosmological constant. He hoped and felt that in time, physics and astronomy would be able to allow the cosmological constant to go away. When Hubble found the expansion of the universe and the Big Bang theory were proposed, Einstein thought that his cosmological constant was his biggest blunder. But was it?
Now, with the introduction of dark energy to help explain the expansion of the universe, the cosmological constant was reintroduced. As explained last time, if the dark energy density is constant, the universe will be open and expand forever. With a constant dark energy density, this implies that the universe is homogeneous in both space and time. Remember that this is referred to the Perfect Cosmological Principle which was briefly mentioned here. In other words, the universe appears static and therefore, the cosmological constant may be a physical quantity describing the dark energy density of the universe. Unfortunately, we still don't know what the dark energy density is doing and it may be centuries or millennia before we know.
Showing posts with label General relativity. Show all posts
Showing posts with label General relativity. Show all posts
10 March 2015
05 February 2015
Special Relativity
Previously, we briefly discussed General Relativity and how
it expands Newtonian Mechanics to include the interaction between mass and
space(time). Today, we are going to talk a little bit about Special Relativity
in that what happens to physics when we approach velocities near the speed of
light.
There are two postulates that Albert
Einstein proposed make up Special Relativity.
1.
All Laws of Physics are invariant in
all inertial systems - basically, the laws of physics must remain the same for
all reference frames that are not accelerating.
2.
The speed of light in a vacuum is
the same for all observers, regardless of the motion of the observers. In other
words, the speed of light, c, is the same for a stationary person and a person
moving at any speed, up to the speed of light (which is impossible).
One of the amazing things about Special Relativity is the
idea of time dilation. It means that for any body travelling at any speed, time
is not constant. To a stationary observer, it would appear that the clock for
the moving object slows down, but for the observer in motion, the clock of the
stationary person speeds up. In both cases, each observer sees the clock in his
or her reference frame as moving normally. How does this work?
We have to use math to show this. There is an equation that
describes how time is relative using the Lorentz transformation:
Where 
is the time for the moving observer, 
is the time for the stationary observer, and 
is the Lorentz factor given by 
,
where v is speed of the moving
observer and c is the speed of light
(3x108 m/s). Since v is
always less than the speed of light, is
always greater than one, therefore Δt’
> Δt. For the stationary observer,
the clock of the moving person is slow, while for the moving observer, the
stationary person has a fast clock. For velocities much smaller than the speed
of light, γ is virtually 1 and the
times are equivalent. However, we have seen for objects moving near the speed
of light (particles with little mass), this actually holds true. An example is
a muon. A stationary muon would decay in only 2.2 μsec; however, when a muon is
travelling near the speed of light will last much longer than 2.2 μsec (to a
stationary observer).
is the time for the moving observer, is the time for the stationary observer, and is the Lorentz factor given by ,
where v is speed of the moving
observer and c is the speed of light
(3x108 m/s). Since v is
always less than the speed of light, is
always greater than one, therefore Δt’
> Δt. For the stationary observer,
the clock of the moving person is slow, while for the moving observer, the
stationary person has a fast clock. For velocities much smaller than the speed
of light, γ is virtually 1 and the
times are equivalent. However, we have seen for objects moving near the speed
of light (particles with little mass), this actually holds true. An example is
a muon. A stationary muon would decay in only 2.2 μsec; however, when a muon is
travelling near the speed of light will last much longer than 2.2 μsec (to a
stationary observer).
Another strange feature of special relativity is the idea of
length contraction. In other words, a moving object will appear shorter than it
would be if it were stationary.
Where Δx’ is the length
of the moving object and Δx is the
length of the stationary object. This again leads to another strange phenomenon:
relativity of simultaneity.
Relativity of simultaneity means that something that happens
simultaneously in one inertial frame is not necessarily simultaneous in
another. The best example given is called the ladder paradox.
Imagine a ladder and a barn. The ladder is just a little bit
longer than the length of the barn. Now imagine that the ladder is moving at a relativistic
speed. Someone in the barn reference frame will see the ladder shorter than its
stationary length, and can close doors at both ends with ladder inside
simultaneously. However, to keep the ladder from crashing through the end door,
the doors must open up again. For the ladder, however, it sees the end door
close first, then the front door close secondly, with the end door opening up
before the ladder reaches it. The ladder does not see both doors closed at the
same time.
A third phenomenon of special relativity is the idea of
infinite mass. For a moving mass, Δm’=γΔm.
As v approaches c, you can see that γ
nears infinity (v/c approaches 1 and
the denominator in the Lorentz factor approaches 0). This is why nothing can
reach the speed of light as its mass will become infinite, which leads to
another famous equation: E=mc2, the energy-mass equivalency. This
means in order to accelerate a mass to the speed of light, the amount of energy
required goes to infinity.
04 February 2015
General Relativity and Astronomy
Previously, we discussed how mass can curve space(time) due to general relativity. Why is this important?
The curvature of mass leads to interesting phenomena. The first is it causes the perihelion of a planet orbiting the Sun to precess. Secondly, it causes light actually to bend - yes, gravity affects light - and this leads to really weird stuff.
Let's look at the first one. The best example of the precession of a planet at it orbits the Sun is the path of Mercury. This was discussed back in the post about Mercury and General Relativity. The highlight of the discussion was that as Mercury orbits around the Sun, at perihelion, Mercury is in the deepest part of the gravity well created by the Sun. As it continues to orbit, each successive perihelion moves farther ahead in its orbit. The perihelion of Mercury was noticed in the mid 1800s, but was thought to be caused by an inner planet. But after Einstein's Theory of General Relativity was developed, the equations were able to show why Mercury's orbit precessed around the Sun. Everything that orbits around another body shows this precession, with the amount of precession dependent on the mass of the central body.
The second one seems a little weirder. From Newton's equation for universal gravitation, we see that the force of gravity is dependent on mass. However, light and all electromagnetic radiation are massless. So how does gravity bend light?
In a nutshell - gravity wells.
If a star were behind the Sun, in Newtonian gravity, we would not be able to see it, because gravity only affects objects with mass. We would see something like this.
However, because of General Relativity, light will bend in the presence of a gravitational field. The light from the star will curve around the Sun and we can see it from Earth.
This was actually proven by Sir Arthur Eddington. In 1919, there was a solar eclipse that he observed and photographed. When the pictures were analyzed, they could see the effect of gravity on stars. This analysis only worked during an eclipse because otherwise, the Sun would be too bright and wash out the background stars. Eddington gave physical proof that General Relativity was correct!
Positive and Negative Image of Solar
Eclipse of May 1919
Image Credit:
This phenomena of light bending around masses also is used to search for exoplanets. As a planet passes in front of star, that planet can bend the light towards us. This process is called lensing. Not only does it allow the light to reach us even if the source is behind the mass, it can also magnify the light, making it brighter. Most of the gravitational lensing seen are galactic in nature.
The curved arcs are the lensed
(background) object. The centers are the lenses bending the light.
Image Credit:
Einstein's Cross (Gravitation
Lensing) - Quasar being lensed by a central dim galaxy
Image Credit:
Einstein Ring - When the background
object is perfectly aligned with lensing object and the Earth, a complete ring
can be created
Image Credit:
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