# Presentation: 407707

## Relativity  

- Ricardo Eusebi
**Fermi National Accelerator Laboratory**

_*Outline :*_
*** ******Introduction***
*** ******Special Relativity***
***Summary***
*** ******General Relativity***
***Summary***

## Intuition and common sense in Relativity

- What happens if we just increase the velocity of the train to, say V=3⁄4c ?

**V = 1 km/sec **

**     ****distance=0       d=V x  5 sec       d=V x 10 sec**        **d=V x 15 sec** **            d = V x time**

- You on land, friend driving train. Tell you friend to turn the lights on 10 seconds after passing stop A.

_**A**_**                                         **_**B**_

**V = 225,000 km/sec **

**     ****distance=0       d=V x  5 sec       d=V x 10 sec**        **d=V x 15 sec** **            d = V x time**

_**A’**_**                                         **_**B’**_

- This effect was theoretically predicted before any experiment confirmed it!!
- To understand this we need to go back to the times of Galileo

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Physics before Einstein

- Ricardo Eusebi - Saturday Morning Physics

## Principle of Inertia

- Deduced laws of planetary movement.
-  But why planets keep on moving ?
- “*Angels are beating their wings and pushing them.*”

- Galileo: “*Nothing pushes the planets in their orbits*”
*An object in uniform motion or at rest remains in a state of uniform motion or at rest unless acted upon by a force.*

- Kepler

- Galileo

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Principle of Relativity

***Principle of Relativity****: ****All states of uniform motion are equivalent for the description of physical phenomena***

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Galilean Transformations

**Transformation:** Allows a measurement of time and position in one coordinate system to be translated to another coordinate system.

- me

- you

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## What if the reference frames are moving ?

- Let’s synchronize the reference frames such that they were on top of each other at time t=0
- Now the distance between the two frames (before a constant 7 miles) grow with time as V x t

- me

- you

- V

- This is a Galilean
- Transformation

- From now on let’s refer to the moving frame with primed coordinates x’ and t’

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Galilean Transf + Relativity Principle

***Galilean trans: ***transformation rules between inertial reference frames  (r.f.).
***Relativity Principle: ***Laws of Physics  invariant to transformation of inertial r.f.’s.
**We can test this using the Galilean transformations!**
- 	If a Physics law, say F(x’,t’)=b, is valid in S’,  law  F(x,t)=b must be valid in S
**Examples:**
- Applying the Galilean transform to a velocity **v’** we obtain:
- What about acceleration, how does **a’** relates to **a** ?

- Transformation of velocities

- Accelerations are the same

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Galilean Transf + Relativity Principle

- What about Newton’s law ?
- Gravitational force between objects with masses m1 and m2 ?
- Wave equations: any function (x,t) that satisfy this is a Wave

- Newton’s law is invariant

- Grav. force is invariant

- Waves are **NOT** invariant to galilean transformations!
- What looks like a wave in one r.f. does not look like a wave in another.

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Before 1900’s

**Situation before Einstein **
- Physics independent of inertial r.f. All inertial r.f.’s are equivalent.
- Galilean transformations translate from one frame to another
- Velocities could go up to infinite: v’=v-V

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## A Problem Arises

- Ricardo Eusebi - Saturday Morning Physics

## The problem: electromagnetism (light)

**Electromagnetic waves**
- Maxwell wrote the basic theory in 1860’s
- Heinrich Hertz sent and received radio waves in 1890’s
- Radio waves predicted and measured to move with the speed of light!
**Problem:**
**Maxwell equations are not invariant to Galilean transformations of r.f.**
**Three options:**
- Maxwell equations are wrong.
- Galilean transformations are wrong.
- The relativity principle is wrong for light.
- 	Is there an ether through which light propagates ?

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## On the question of the ether

- Michelson-Morley experiment to detect movement relative to the ether.

- B

- C

- A

- The Michelson-Morley experiment did not detect any variation of the velocity of light. Not in any direction, nor in any day of the year.

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## The solution, by Albert Einstein

- Ricardo Eusebi - Saturday Morning Physics

## Enter Einstein

**Three options:**
- Maxwell equations are wrong.
- Galilean transformations are wrong.
- The relativity principle is wrong for light.
- 	Is there an ether through which light propagates ?
**He postulated:**
_*Principle of relativity is valid.*_
- 	The principle of Relativity is symmetrically nice. Only ether could violate it, but the ether has not been found.
_*The velocity of light (in vacuum) is the same in all r.f.’s *_
- 	Maxwell equations predict the velocity of light, which has been measured to be the same as predicted. Maxwell is right and the velocity of light is the same regardless the r.f. from which we look.

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Einstein Postulates

- Today we can experimentally test these postulates
- No law has ever been seen different, in different reference frames.
- Constancy of the velocity of light in any frame is **VERIFIED**!

- v

- c+v ?

- c-v ?

- c !

- c !

- 0.01c ?

- c !

- 1.99c ?

- c !

_**In Pion  decays**_

-  _**In double stars**_

- 

- 

- 

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Time in Relativity

- Standing in a moving r.f. S’ we make a clock based on the velocity of light.

- d/2

- t’=d/c

- d/2

- d/2

- ct/2

- ct/2

- V=c

- From the  r.f. S we may see this differently

**Moving clocks run slower than to non-moving clocks !!!**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Space in Relativity

- Standing in a moving r.f. S’ we measure L’ based on the time.

- t’=2L’/c

- From the  r.f. S we may see this differently

**Moving Lengths look shorter !!!**

- L’

- L

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Putting it all together

- Galilean transformations are no more:
- _From which we can infer:_
**Length contraction:**
**Time dilation:**
**Addition of velocities:**

- Lorentz transformations have replaced them

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Time dilation is proven every day

- Muons are particles that decay (on average) every 2 s.
- _Muon at rest_
- _Muon moving at V=0.98c__ _

- t’=t’0+2 s

- t’=t’0+1 s

- t’=t’0+3 s

- t’=t’0

- t’=t’0+4 s

- t=t0+10 s

- t=t0+5 s

- t=t0+15 s

- t=t0

- t=t0+20 s

- According to Galilean transformations   if the muon is moving at V=0.98c we’ll see it disintegrating at

- Km

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Time dilation in muons

- _Muon traveling at 0.98c down the atmosphere_

**As viewed in the earth reference frame**
- The landscape has it’s normal size.
- If muons had sizeis flattened like a pancake.
- Muon’s clock run 5 times slower.

- V=0.98c

- L=10Km

**As viewed in the muon’s reference frame**
- If muons had size  muon has  its normal size.
- The  landscape (moving) is flattened like a pancake.
- Muon’s clock runs normally. But distance is 5 times shorter.

- V=0.98c

- L=1.96Km

**Both frames predict the same results: muon decays at the base**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Mass in Relativity

**In the C.M. rest frame**

_**In the Ball “0**__**x**__**” r.f. using:**_
**conservation of momentum in Y**
**Law of addition of vel.: V****y****(0)=V****y****(1)******

**Mass = m(********), the mass depends on the velocity !**

**Ball “0****x****”r.f **

_**Consider two balls colliding symmetrically to each other**_

**0**

**1**

**Ball “1****x****”r.f **

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Energy in Relativity

**We obtained:**
**Replacing that we get:**

- for <<1 we can approximate

**E****tot ****  =  ********m****0****c****2****   = E****rest **** +   E****kin**

**E****tot****=mc****2**
**(it’s m, not m****0****)**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Simultaneity in Relativity

**Definition of *****event*****: **
*Something that happens at a specific place at a specific time*
*Event’s time is measured by a clock at the event’s position.*
**Let consider two events E****1**** and E****2****  **
- E1 has associated a pair x1 and t1
- E2 has associated a pair x2 and t2
**From a r.f moving at V we measure:**

- E1

- E2

- V

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Simultaneity in Relativity

- Assume E1 and E2  are simultaneous (or t1= t2)
- In the moving reference frame:
- Can these two events be related, can one be the cause of the other one ?
**No! There is no signal  that can cover the distance ********X in no time!**

**The two events are not seen as simultaneous!**
**If V>0 (as drawn)  E****2**** happens first, later E****1**
**If V<0 E****1**** happens first, later E****2**
**Event in the direction of the moving r.f. happens first, in this case E****2**

- E1

- E2

- V

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Paradox #1: Car and garage  

- Paradox #1: Car and garage

- Car fits perfectly in garage.

- Garage length L

- Garage

- V

- Garage

- V

- Car driver’s perspective; “*Garage looks shorter, it won’t fit !!!**”*

- Car and garage moving with respect to each other
- Guy in garage’s perspective: ”*Car looks shorter, no problem, it’ll fit in.*”

- d

- Car is set at distance d  coming with velocity V. Garage set to close doors in d/V seconds.

- L/

- L/

- d/

- Who’s right ? The guy in the garage or the driver ?

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Paradox #1: Car and garage, cont   

- Both sentences are right. The car “**will fit**” and “**will not fit**” in the garage.
- What’s the definition of fit ?

- The car fits in the garage if: the car is within the doors when both doors have been closed SIMULTANEOUSLY .
- “Fit” requires the concept of simultaneity.
- Simultaneity is a frame-dependent concept.
- The “**fit**” mentioned by the driver and the garage-guy represent different things!

- This definitely fits!

- This too!

- Does this fit  ?

- This definitely No

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Paradox #1: Car and garage, cont 2  

- Paradox #1: Car and garage, cont 2

- Car fits perfectly in garage.

- Garage length L

- Garage

- V

- Garage

- V

- How does the car driver see the closing of the doors?

- Car and garage moving with respect to each other
- Guy in garage’s perspective: ”*Car looks shorter, no problem, it’ll fit in*.”

- d

- Car is set at distance d  coming with velocity V. Garage set to close doors in d/V seconds.

- L/

- L/

- d/

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Paradox #1: Car and garage with a twist

- Garage

- V

- Car and garage moving with respect to each other from the Car’s perspective.
- Let’s add a twist : the left door is jammed and continuously closed.

**Question :**
- While the car crashes on the left door, will then the right door cut the remaining of the car left outside the garage ?
**NO!. The rear of the car does not know the front crashed until some time later.**

- V

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Twin paradox

**Twin experiment:**
- Twin Earl stays on earth, twin Molly moves in a spacecraft at 0.999c (=22.3)
- Molly eventually reaches a planet 20 light years from earth, as measured from earth. She turns back and heads towards earth back.
- In her clock Molly reaches the earth back 1.8 years older than we she left. According to earth clocks Earl is 40 years older.
**Pseudo-Paradox reasoning: **
- I can always interpret Molly at rest and Earl moving away and back.
- If the movement between these two twins is relative, why should one be older than the other ?
**Solution: **
- You cannot interpret Molly at rest and Earl moving away and back !
- You can definitely determine which went away and which stayed !
- How ?
- The one who left had to accelerate to come back!

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Hints of General Relativity

- Ricardo Eusebi - Saturday Morning Physics

## General Relativity: Principles

- Special relativity is valid only in Inertial reference frames.
- Einstein wasn’t happy with that. He sought to extend the principle of relativity to ALL frames of reference. He noticed :
_**Principle of equivalence**_
- Accelerated frame and frame at rest with a gravitational pull are equivalent in every respect

**=**
- It is impossible to distinguish between accelerated r.f. and the frame at rest with a gravitational pull.

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## General Relativity: Frames

- Consider a free falling reference frame

- Surface of planet

- Surface of planet

- Free  falling reference frames are indistinguishable from inertial reference frames

- =

- =

- Equivalence principle

- Forces cancel each other

**This is an inertial r.f. !!**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Doppler effect

_**Non relativistic Doppler effect:**_
**f’ = f  ( 1 + V/c)**
_**Relativistic Doppler effect:**_
**Include that the moving observer’s clock runs slower  by ******
**f’ = ******** f  ( 1 + V/c)**

**f**

- Light frequency is relative to the observer!

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Measuring star velocities with Doppler effect 

- Measuring star velocities with Doppler effect

**Naked star**
**Star with a gas cloud**

- H

**Absorption lines: fingerprint of cloud**

**Light spectrum**

**Binary system**
**Using f’ = ******** f  ( 1 + V/c)**

- v

**Absorption lines Shifted**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## General Relativity: Clocks in gravity fields

- Let’s consider an **accelerated** frame. From doppler effect
- A photon of frequency *f* is emitted from the floor.
- It is detected in the ceiling with frequency *f’**<**f*

- =

- Equivalence principle

**Clocks close to a massive body run slower than clocks farther away**

**Experimentally verified by many “Tower Experiments”**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## General Relativity: Light in gravity fields

- Let’s consider an **accelerated** frame.
- A photon pass through the frame transversally from left to right.

**Light bends in a gravitational field as would any other particle.**

- =

- Equivalence principle

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Experimental confirmation of light-bending

**Courtesy of “*****Space Telescope Science Institute”***

_**Example:**_
-     	      =  Real object
- 	      =  Apparent object

**galaxy**

**Hubble Telescope**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## General relativity: the bending of space

**A1**

**Is the space curved ?**

**Einstein concluded: Space-time must be curved !**

- _Q: is earth curved ?_

**A1**: Sum internal angles of triangle greater than 180.
**A2:** Circumference c of a circle is less than 2r

- r

- c

- _Q: is space-time curved ?_

**A:** circumf. of circle is not 2r !!

- Shorter by 

**90o**

**90**

**90**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## General Relativity: in a nutshell

**Massive objects curved space-time.**
- The more massive the greater the curvature.
- The greater the curvature the more intense the gravity.
**Objects move following the shortest path in curved space-time.**

- (Taken from Bryan Mendez web page http://cse.ssl.berkeley.edu/bmendez/ay10/2002/notes/lec15.html)

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Gravitational waves: Ongoing experiments

**Many internationals efforts:**
- VIRGO (Italy/France)
- GEO (Germany/Great Britain)
- ACIGA (Australia)
- TAMA (Japan).
**LIGO **(USA) : two Michelson-Morleystyle interferometers.
- LISA: space-based observatory scheduled to launch in 2011.

**LIGO sites****:**

**Louisiana State**

**Gravitational waves:**
-  generated when massive things accelerate in a particular way.
-  ripples in spacetime, spreading  outward like ripples on a pond.

**No direct evidence of gravitational waves yet found.**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Gravitational waves: Indirect evidence

**Gravitational waves:**
- generated when massive things accelerate in a special way.
- ripples in space-time propagating at the velocity of light
- Binary systems must be losing energy if they emit gravitational waves.

**The energy of binary systems diminish as expected from gravitational-wave radiation!**

- Ricardo Eusebi - Saturday Morning Physics

- April 14, 2007

## Thank you very much.Questions ?

- Ricardo Eusebi - Saturday Morning Physics