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# Download (Very) basic introduction to special relativity

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(Very) basic introduction to special relativity Math 3560 Groups and symmetry, Fall 2014 Raul Gomez 1/ 31 1 Newton’s reference frames Newton’s reference frames Figure : Isaac Newton Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 2/ 31 1.1 Newton’s three laws Newton’s three laws 1. Law of inertia An object either is at rest or moves at a constant velocity, unless acted upon by an external force. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 2/ 31 1.1 Newton’s three laws Newton’s three laws 1. Law of inertia An object either is at rest or moves at a constant velocity, unless acted upon by an external force. 2. Law of motion The acceleration of a body is directly proportional to the force acting on the body, and inversely proportional to its mass. In mathematical terms, F = m a. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 2/ 31 1.1 Newton’s three laws Newton’s three laws 1. Law of inertia An object either is at rest or moves at a constant velocity, unless acted upon by an external force. 2. Law of motion The acceleration of a body is directly proportional to the force acting on the body, and inversely proportional to its mass. In mathematical terms, F = m a. 3. Law of action-reaction For every action there is an equal and opposite reaction. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 3/ 31 1.1 Newton’s three laws About Newton’s first law Newton’s first law says that there is a reference frame (that is there is a system of coordinates) on which Newton’s first law holds. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 3/ 31 1.1 Newton’s three laws About Newton’s first law Newton’s first law says that there is a reference frame (that is there is a system of coordinates) on which Newton’s first law holds. Any such a frame is called an inertial frame. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 3/ 31 1.1 Newton’s three laws About Newton’s first law Newton’s first law says that there is a reference frame (that is there is a system of coordinates) on which Newton’s first law holds. Any such a frame is called an inertial frame. Although this statement may seem obvious, it is not clear at all how to choose such a reference frame in the universe. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 4/ 31 1.1 Newton’s three laws Example If we use a fixed point on earth to define a reference frame, then we would obtain an example of a non inertial frame. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 4/ 31 1.1 Newton’s three laws Example If we use a fixed point on earth to define a reference frame, then we would obtain an example of a non inertial frame. In this frame the centrifugal force and the Coriolis force (which is responsible for the formation of hurricanes) violate Newton’s first law. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 5/ 31 1.2 Systems of particles Systems of particles Now assume that we have a system of N particles with position ri and mass mi . Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 5/ 31 1.2 Systems of particles Systems of particles Now assume that we have a system of N particles with position ri and mass mi . Then Newton’s second law reads: Fi = mi ai . Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 5/ 31 1.2 Systems of particles Systems of particles Now assume that we have a system of N particles with position ri and mass mi . Then Newton’s second law reads: Fi = mi ai . Decompose Fi as Fi = ∑ Fi j + Fext i , j6=i where Fi j is the force on the i-th particle due to the j-th particle. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames Decompose Fi as 6/ 31 1.2 Systems of particles Fi = ∑ Fi j + Fext i , j6=i where Fi j is the force on the i-th particle due to the j-th particle. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 6/ 31 1.2 Systems of particles Decompose Fi as Fi = ∑ Fi j + Fext i , j6=i where Fi j is the force on the i-th particle due to the j-th particle. Suming over all the i’s we get ∑ Fi = i ∑ i, j, j6=i = i ∑ (Fi j + F ji) + ∑ Fext i . i< j Intro to special relativity Fi j + ∑ Fext i i Raul Gomez ([email protected]) 1 Newton’s reference frames 6/ 31 1.2 Systems of particles Decompose Fi as Fi = ∑ Fi j + Fext i , j6=i where Fi j is the force on the i-th particle due to the j-th particle. Suming over all the i’s we get = ∑ Fi i ∑ i, j, j6=i = Fi j + ∑ Fext i i ∑ (Fi j + F ji) + ∑ Fext i . i< j i But according to Newton’s third law, Fi j = −F ji , hence =: Fext , ∑ Fi = ∑ Fext i i i where Fext is the total exterior force acting on the system. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 1.2 Systems of particles 7/ 31 We define the total mass of the system to be M = ∑ mi , i and the center of mass to be R= Intro to special relativity ∑i mi ri . M Raul Gomez ([email protected]) 1 Newton’s reference frames 7/ 31 1.2 Systems of particles We define the total mass of the system to be M = ∑ mi , i and the center of mass to be R= ∑i mi ri . M Then, if A is the second derivative of R with respect to time, we have that Fext = MA. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 7/ 31 1.2 Systems of particles We define the total mass of the system to be M = ∑ mi , i and the center of mass to be R= ∑i mi ri . M Then, if A is the second derivative of R with respect to time, we have that Fext = MA. In a system where there are no external forces (like the universe) this equation says that fixing the origin at the center of mass gives you an inertial reference frame. Intro to special relativity Raul Gomez ([email protected]) 1 Newton’s reference frames 8/ 31 1.2 Systems of particles Example Let r1 = (0, 0) r2 = (3, 6) r3 = (8, 2), and assume that F12 = (1, 2) F13 = (2, 1/2) F23 = (0, 0). The we obtain the following picture: u r2 F21 9 F12 r1 Intro to special relativity : u u r3 F31 F13 Raul Gomez ([email protected]) 1 Newton’s reference frames 1.3 Absolute vs. relative point of view 9/ 31 Absolute vs. relative point of view Figure : Gottfried Leibniz Intro to special relativity Raul Gomez ([email protected]) 10/ 31 2 Classical Electromagnetism Classical Electromagnetism Figure : James Clerk Maxwell Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 11/ 31 2.1 Maxwell equations Maxwell equations (1862) Let E, B and J be the electric field, magnetic field and current density, respectively, and let ρ be a charge distribution in R3 . 1. Gauss law. ∇ · E = ρ/ε0 . 2. Gauss law for magnetism. ∇ · B = 0. 3. Fadaray Law. ∂B . ∂t 4. Ampère circulation law (with Maxwell correction.) ∂E . ∇ × B = µ0 J + ε0 ∂t ∇×E = − where ε0 and µ0 are the electric and magnetic constants. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 12/ 31 2.1 Maxwell equations Maxwell correction to Faraday’s law is critical in that it predicts the existance of electromagnetic waves. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 12/ 31 2.1 Maxwell equations Maxwell correction to Faraday’s law is critical in that it predicts the existance of electromagnetic waves. More precisely, if ρ = 0, then Maxwell equations are equivalent to the system of equations: ∂ 2E ∂ 2E ∂ 2E ∂ 2E + 2 + 2 − 2 =0 ∂ x2 ∂y ∂z ∂t and ∂ 2B ∂ 2B ∂ 2B ∂ 2B + 2 + 2 − 2 = 0, ∂ x2 ∂y ∂z ∂t plus a condition relating E and B. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 13/ 31 2.2 Electromagnetic waves Electromagnetic waves Maxwell was able to solve his equations in the vacuum, and predicted the existance of electromagnetic waves. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 13/ 31 2.2 Electromagnetic waves Electromagnetic waves Maxwell was able to solve his equations in the vacuum, and predicted the existance of electromagnetic waves. In particular, he was able to calculate the speed at which his predicted electromagnetic waves should travel. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 13/ 31 2.2 Electromagnetic waves Electromagnetic waves Maxwell was able to solve his equations in the vacuum, and predicted the existance of electromagnetic waves. In particular, he was able to calculate the speed at which his predicted electromagnetic waves should travel. The value Maxwell obtained, using the known values at the time of the constants ε0 and µ0 , was essentially the same as the speed of light as measured at that time. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 13/ 31 2.2 Electromagnetic waves Electromagnetic waves Maxwell was able to solve his equations in the vacuum, and predicted the existance of electromagnetic waves. In particular, he was able to calculate the speed at which his predicted electromagnetic waves should travel. The value Maxwell obtained, using the known values at the time of the constants ε0 and µ0 , was essentially the same as the speed of light as measured at that time. Based on this observation, Maxwell predicted that light was just an example of an electromagnetic wave. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 14/ 31 2.2 Electromagnetic waves Maxwell predictions created considerable problems ammong the physics community of the time. The main issues were: Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 14/ 31 2.2 Electromagnetic waves Maxwell predictions created considerable problems ammong the physics community of the time. The main issues were: 1. On earth, waves always propagate through a medium, they can’t exist in the vacuum. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 14/ 31 2.2 Electromagnetic waves Maxwell predictions created considerable problems ammong the physics community of the time. The main issues were: 1. On earth, waves always propagate through a medium, they can’t exist in the vacuum. 2. Maxwell calculated the speed of the electromagnetic waves, but this was the speed with respect to what exactly? Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 14/ 31 2.2 Electromagnetic waves Maxwell predictions created considerable problems ammong the physics community of the time. The main issues were: 1. On earth, waves always propagate through a medium, they can’t exist in the vacuum. 2. Maxwell calculated the speed of the electromagnetic waves, but this was the speed with respect to what exactly? Trying to kill two birds with one stone, they proposed the existance of an essentially indetectible substance called luminiferous aether. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 14/ 31 2.2 Electromagnetic waves Maxwell predictions created considerable problems ammong the physics community of the time. The main issues were: 1. On earth, waves always propagate through a medium, they can’t exist in the vacuum. 2. Maxwell calculated the speed of the electromagnetic waves, but this was the speed with respect to what exactly? Trying to kill two birds with one stone, they proposed the existance of an essentially indetectible substance called luminiferous aether. This was supposed to be the medium through which electromagnetic waves propagated. The speed calculated by Maxwell was then the speed of light with respect to this medium. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 15/ 31 2.3 The Michelson Morley experiment The Michelson Morley experiment So in 1887, Albert Michelson and Edwartd Morley designed an experiment to measure the velocity of the earth with respect to the surrounding aether. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 15/ 31 2.3 The Michelson Morley experiment The Michelson Morley experiment So in 1887, Albert Michelson and Edwartd Morley designed an experiment to measure the velocity of the earth with respect to the surrounding aether. Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 15/ 31 2.3 The Michelson Morley experiment The Michelson Morley experiment So in 1887, Albert Michelson and Edwartd Morley designed an experiment to measure the velocity of the earth with respect to the surrounding aether. The result however, was totally unexpected. The speed of light was the same in every direction! Intro to special relativity Raul Gomez ([email protected]) 2 Classical Electromagnetism 2.3 The Michelson Morley experiment 16/ 31 Figure : Albert Einstein Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 17/ 31 3.1 Space-time coordinates Space-time coordinates To simplify calculations, from now on we will only consider universes with one spatial dimension. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 17/ 31 3.1 Space-time coordinates Space-time coordinates To simplify calculations, from now on we will only consider universes with one spatial dimension. That is, we will consider events in the universe by a space coordinate and a time coordinate. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 17/ 31 3.1 Space-time coordinates Space-time coordinates To simplify calculations, from now on we will only consider universes with one spatial dimension. That is, we will consider events in the universe by a space coordinate and a time coordinate. Observe that if r(t) represents the position of a particle at time t, then we can describe its position as a point in R2 using the coordinates (t, r(t)). Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 3.2 The Minkowski metric 18/ 31 The Minkowski metric Figure : Hermann Minkowski Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 19/ 31 3.2 The Minkowski metric The Minkowski metric Is it possible to define a concept of distance between different events in the universe? Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 19/ 31 3.2 The Minkowski metric The Minkowski metric Is it possible to define a concept of distance between different events in the universe? For example, we can say:“The homecoming fireworks were lighted up last Friday at Schoellkopf Field.” Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 19/ 31 3.2 The Minkowski metric The Minkowski metric Is it possible to define a concept of distance between different events in the universe? For example, we can say:“The homecoming fireworks were lighted up last Friday at Schoellkopf Field.” To express this event in coordinates we would need three spacial coordinates and one temporal coordinate. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 19/ 31 3.2 The Minkowski metric The Minkowski metric Is it possible to define a concept of distance between different events in the universe? For example, we can say:“The homecoming fireworks were lighted up last Friday at Schoellkopf Field.” To express this event in coordinates we would need three spacial coordinates and one temporal coordinate. But if we ask about the distance between this moment and the homecoming event, we only need two numbers: Schoellkopf Field is half a mile from here, and it’s been four days since Friday. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 19/ 31 3.2 The Minkowski metric The Minkowski metric Is it possible to define a concept of distance between different events in the universe? For example, we can say:“The homecoming fireworks were lighted up last Friday at Schoellkopf Field.” To express this event in coordinates we would need three spacial coordinates and one temporal coordinate. But if we ask about the distance between this moment and the homecoming event, we only need two numbers: Schoellkopf Field is half a mile from here, and it’s been four days since Friday. Can we combine this 2 numbers to get a notion of “distance” between this two events? Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 20/ 31 3.2 The Minkowski metric It may seem natural to simply consider the time as another coordinate and use the Pythagorean theorem to compute the “distance” between events. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 20/ 31 3.2 The Minkowski metric It may seem natural to simply consider the time as another coordinate and use the Pythagorean theorem to compute the “distance” between events. Unfortunately this is not very useful because the laws of physics are not invariant under the action of SO(4, R). Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 20/ 31 3.2 The Minkowski metric It may seem natural to simply consider the time as another coordinate and use the Pythagorean theorem to compute the “distance” between events. Unfortunately this is not very useful because the laws of physics are not invariant under the action of SO(4, R). For example, if we write Maxwell equations and then we transform everything using an element of SO(4, R) we get an equation that looks very different. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 21/ 31 3.2 The Minkowski metric This is clear if we look at the wave equation in one dimension: ∂ 2u ∂ 2u − = 0. ∂ x2 ∂t 2 Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 21/ 31 3.2 The Minkowski metric This is clear if we look at the wave equation in one dimension: ∂ 2u ∂ 2u − = 0. ∂ x2 ∂t 2 Looking at this equation suggest a different way of calculating the “distance” between 2 events. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 21/ 31 3.2 The Minkowski metric This is clear if we look at the wave equation in one dimension: ∂ 2u ∂ 2u − = 0. ∂ x2 ∂t 2 Looking at this equation suggest a different way of calculating the “distance” between 2 events. Definition Let v1 = (t1 , x1 ) and v2 = (t2 , x2 ) be two vectors in R2 . We define the Minkowski product of v1 and v2 by v1 ◦ v2 = t1t2 − x1 x2 . Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 22/ 31 3.2 The Minkowski metric Definition Let v1 = (t1 , x1 ) and v2 = (t2 , x2 ) be two vectors in R2 . We define the Minkowski product of v1 and v2 by v1 ◦ v2 = t1t2 − x1 x2 . Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 22/ 31 3.2 The Minkowski metric Definition Let v1 = (t1 , x1 ) and v2 = (t2 , x2 ) be two vectors in R2 . We define the Minkowski product of v1 and v2 by v1 ◦ v2 = t1t2 − x1 x2 . Using this inner product, we can define a new “norm” (or rather its square) by N(v)2 = v ◦ v, Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 22/ 31 3.2 The Minkowski metric Definition Let v1 = (t1 , x1 ) and v2 = (t2 , x2 ) be two vectors in R2 . We define the Minkowski product of v1 and v2 by v1 ◦ v2 = t1t2 − x1 x2 . Using this inner product, we can define a new “norm” (or rather its square) by N(v)2 = v ◦ v, and using this norm we can define the Minkowski “metric” d(v1 , v2 )2 = N(v1 − v2 )2 . Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 23/ 31 3.2 The Minkowski metric and using this norm we can define the Minkowski “metric” d(v1 , v2 )2 = N(v1 − v2 )2 . Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 23/ 31 3.2 The Minkowski metric and using this norm we can define the Minkowski “metric” d(v1 , v2 )2 = N(v1 − v2 )2 . The name “metric” here is a little misleading because the number I’m getting on the right may very well be negative, in which case, taking the square root is not really well defined. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 23/ 31 3.2 The Minkowski metric and using this norm we can define the Minkowski “metric” d(v1 , v2 )2 = N(v1 − v2 )2 . The name “metric” here is a little misleading because the number I’m getting on the right may very well be negative, in which case, taking the square root is not really well defined. However the Minkowski product of a vector in R2 with itself is well defined and is a very useful quantity as we will see soon. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 24/ 31 3.3 Special relativity Special relativity The point of the Minkowski inner product is that Maxwell equations (and in general the equations of physics) are preserved under the action of SO(1, 1). Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 24/ 31 3.3 Special relativity Special relativity The point of the Minkowski inner product is that Maxwell equations (and in general the equations of physics) are preserved under the action of SO(1, 1). That is, I can use SO(1, 1) to change my reference frame and still get the same physics laws. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 24/ 31 3.3 Special relativity Special relativity The point of the Minkowski inner product is that Maxwell equations (and in general the equations of physics) are preserved under the action of SO(1, 1). That is, I can use SO(1, 1) to change my reference frame and still get the same physics laws. As we will see shortly, this is extremely useful. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 25/ 31 3.3 Special relativity Let’s start by considering the points in R2 such that v ◦ v = 0. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 25/ 31 3.3 Special relativity Let’s start by considering the points in R2 such that v ◦ v = 0. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 25/ 31 3.3 Special relativity Let’s start by considering the points in R2 such that v ◦ v = 0. This are called the light rays, precisely because they describe the trajectory of light coming up from the origin. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 26/ 31 3.3 Special relativity The rest of the plane gets divided into two regions, the time-like directions: {v ∈ R2 | v ◦ v > 0}, and the space-like directions: {v ∈ R2 | v ◦ v < 0}. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 27/ 31 3.3 Special relativity Observe now that the set {v ∈ R2 | v ◦ v = 1}, describes an hyperbola. This are the points that, in a sense, are at “distance” 1 from the origin. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 27/ 31 3.3 Special relativity Observe now that the set {v ∈ R2 | v ◦ v = 1}, describes an hyperbola. This are the points that, in a sense, are at “distance” 1 from the origin. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 28/ 31 3.3 Special relativity Now we will consider two particles, r1 and r2 traveling at constant speed from the origin and whose graph in R2 are given by the parametric equations: t 7→ (t, 0), t 7→ (t cosh α,t sinh α). Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 3.3 Special relativity 29/ 31 If we transform this picture using the element cosh α − sinh α g(−α) = , − sinh α cosh α we obtain: Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 3.3 Special relativity 29/ 31 If we transform this picture using the element cosh α − sinh α g(−α) = , − sinh α cosh α we obtain: Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 29/ 31 3.3 Special relativity If we transform this picture using the element cosh α − sinh α g(−α) = , − sinh α cosh α we obtain: Observe that this means that for both observers their relative velocities are the same, and the speed of light is the same with respect to both observers! Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 3.3 Special relativity 30/ 31 How can this be possible? Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 30/ 31 3.3 Special relativity How can this be possible? This is because the events that occur at the “same time” depend on the observer! Intro to special relativity Raul Gomez ([email protected]ll.edu) 3 Special relativity 30/ 31 3.3 Special relativity How can this be possible? This is because the events that occur at the “same time” depend on the observer! Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 30/ 31 3.3 Special relativity How can this be possible? This is because the events that occur at the “same time” depend on the observer! Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 31/ 31 3.3 Special relativity We will try to ilustrate this phenomenom with the following example: Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 31/ 31 3.3 Special relativity We will try to ilustrate this phenomenom with the following example: Suppose that we have an observer that is traveling inside a ship, and we have a second observer watching him through a window in the coast. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 31/ 31 3.3 Special relativity We will try to ilustrate this phenomenom with the following example: Suppose that we have an observer that is traveling inside a ship, and we have a second observer watching him through a window in the coast. There is a lamp in the middle of the room, and observer 1 turns the lamp on. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 31/ 31 3.3 Special relativity We will try to ilustrate this phenomenom with the following example: Suppose that we have an observer that is traveling inside a ship, and we have a second observer watching him through a window in the coast. There is a lamp in the middle of the room, and observer 1 turns the lamp on. For the first observer the light will reach all the walls in the room at the same time. Intro to special relativity Raul Gomez ([email protected]) 3 Special relativity 31/ 31 3.3 Special relativity We will try to ilustrate this phenomenom with the following example: Suppose that we have an observer that is traveling inside a ship, and we have a second observer watching him through a window in the coast. There is a lamp in the middle of the room, and observer 1 turns the lamp on. For the first observer the light will reach all the walls in the room at the same time. However, the observer at the coast will see the light arriving to the different walls at different times. Intro to special relativity Raul Gomez ([email protected])