Tag Archives: spacetime manifold

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July 8, 2015

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Special relativity in Minkowski graph (II)

In the previous blog post we noted that the Lorentz transformation is the map from one inertial coordinate system to another. This means that the coordinates of a person standing on the platform can be mapped to the coordinates of the person sitting in the train using the transformation, which results from special relativity postulates, and quantitative analysis of the physical phenomenon can be done in both the frames.

We also note that these inertial frames (coordinate systems) are same in nature (that is the reason why we used the same constant c in both the coordinate systems). In other words, no inertial frame is to be given any sort of preference. If you like, you can think a 4 dimensional grid of space and time built from the 4 dimensional version of a unit cube. The 2 dimensional version of this grid is shown in the figure of coordinate chart below.

Now this grid must be exactly same for all the coordinate systems (henceforth, we will talk of inertial coordinate systems only). Therefore the grids in coordinate charts 1, 2 and 3 (of different inertial observers) are congruent. All that differs is the mapping of same spacetime events. For instance the green dots represent the events of aging of Bob. Now this set of spacetime events is mapped differently in different frames as shown.

More precisely, one meter “of” a coordinate system should be equal to one meter “of” any other coordinate system and so must be the case with one second (I am really not sure which preposition to use here, but I think “of” should convey the message). These units of measure which define the grid size, are same for all the frames and are termed as “proper length” and “proper time”. Although, a rod (a set of special spacetime events defined later) of one meter in one frame might not be one meter in other coordinate system. In fact Lorentz transformation makes sure that the length is contracted by a factor of \gamma = 1/\sqrt{1-\frac{v^2}{c^2}} (https://en.wikipedia.org/wiki/Length_contraction). We will also explain this phenomenon later in this post.

The German mathematician Hermann Minkowski utilized the property of coordinate charts (maps from manifold to \mathbb{R}^4) in making a useful geometrical tool called Minkowski graph. The idea is that on a graph paper we have two dimensions at our disposal. And, the Lorentz transformation shows that if relative velocity between two frames is along x-axis then the map of the coordinates y and z to another coordinate system is identity. Therefore we consider the map from the events in the spacetime manifold to \mathbb{R}^2 space (our graph paper) which includes only two coordinates x and t as shown in the figure.

chart

In this coordinate system (frame of reference), the event A has been assigned the coordinates (5,3). Also note that the time axis has been divided by the constant c and, here, we define one light second as the time taken by light to travel one meter (or inverse of c).

Now each point in this graph (coordinate system) represents an event in the spacetime manifold. As the consequence, the evolution of a particle in the spacetime is mapped to a trajectory in the graph (which is drawn by the observer at the origin). This trajectory is called the “world line” of the particle. On carefully examining the transformations, one can note that when the relative speed between two frames is greater than the constant c (which we now set to unity) then the factor \gamma becomes imaginary. Furthermore, the factor approaches infinity when the relative speed approaches unity. In relativistic dynamics the total energy of a free massive particle is directly proportional to \gamma. Hence it would require infinite amount of energy to accelerate a particle near the speed of light.

The slope of the trajectory in the Minkowski graph is the inverse of the speed of particle with respect to the observer (who is drawing the graph). Therefore the slope of the trajectory is always greater than unity in the graph because a massive particle cannot be accelerated to the speed of light. A light ray always follows the trajectory of straight line with unity slope. The Lorentz transformation makes sure that the trajectory of the light remains same in all the graphs, corresponding to various coordinate systems.

The Minkowski graph is quite helpful in comparing the physics from two frames. Consider two observers F and F’ moving with some relative speed.

Lorentz_transforms_2.svg-2

Observer F uses her coordinate system to draw a Minkowski graph as shown in the figure above.

Now consider a massive particle at the origin with respect to the observer F’ (i.e x^\prime =0) from time t^\prime = 0 to t^\prime = 1. So how will F draw the world line of the particle in her coordinate system (or Minkowski graph)? As you might have guessed, we will use the Lorentz transformation (because we have to map the information given in F’ to F). In other words we have to find the equation of the particle in coordinates of F subjected to the constraints \Delta x^\prime = 0 and \Delta t^\prime = 1 in the coordinates of F’.

From the Lorentz transformation, the first constraint will give a trejectory equation \Delta t = \frac{\Delta x}{v} which F will interpret as the time axis of the observer F’ (because the line of constant x-coordinate giving t-axis and vice versa is a property of the orthogonal coordinate system in the \mathbb{R}^2 space). Therefore F will draw a straight line in her graph with slope as inverse of the relative speed. Let v=.6 meters per light second. The question now arises is that how will she mark the scale of that time axis in her coordinate system (or graph)? The second constraint gives a result that \Delta t = \frac{1}{\sqrt{1-v^2}} which means that a unit second in the coordinates of F’ is mapped t0 1.25 seconds in the coordinates of F. It means that the two events with a unit time separation at the origin in frame F’, are separated by 1.25 seconds in the frame F. Thus the graph of F will look like

minkowskitime

The violet dots correspond to the events with unit time intervals and zero space displacements in F’ (hence it is the time axis for F’) while green dots in vertical line correspond to the events with unit time intervals and zero space displacements in F. Note that in the graph drawn by the observer F, the 15 units of her time is equal to the 12 units of the time in F’. It is consistent with the fact that \Delta t = 1.25\Delta t^\prime. This is why we say that moving clocks are slower.

Similarly, we can draw the x-axis of F’ in the coordinates of F using the same technique. Consider a situation in which \Delta t^\prime = 0 (simultaneous events) and \Delta x^\prime = 1. Using the Lorentz transformation and first constraint we get \Delta t = v\Delta x. This equation gives the x-axis of the observer F’ in the frame of F. Second constraint gives the relation \Delta x = \frac{1}{\sqrt{1-v^2}}. This means that a unit length (between two simultaneous events) in F’ is mapped to 1.25 meters in F. The graph now looks like

minkowskispacetimeThe violet line with lesser slope represents all the simultaneous events of F’ (with t^\prime = 0) in the coordinates of F where they are having different coordinate t. Thus simultaneity is a relative concept and depends of the frame of reference.

In the end of this post, we will explain length contraction using the Minkowski graph. The length of a rigid rod is defined as the distance (in \mathbb{R}^n space) between its end points at the same time coordinate (convince yourself!). Consider a rod of 2 meters length at rest in the frame F’. Let its first end point be at (4, t^\prime). Therefore, according to the definition of length, the coordinates of other end point will be (6, t^\prime). Now this rod will trace a “world sheet” in the Minkowski graph as shown in the figure below. The red dots on the lower violet line (the x-axis of F’) are the coordinates of the endpoints of the rod which are, according to F’, (4,0) and (6,0). They evolve in the time t^\prime such that after unit second in F’, the coordinates are (4,1) and (6,1) (the couple of next red points on each red line) in F’. And it should be, because the 2 meter rod is at rest in F’.

minklencont

Now how much length will the observer F measure? Let us say that at t=8 light seconds she measures the length. Now according to the definition of the length, she will cut the world sheet of the rod with a line (to obtain simultaneous coordinates in her frame) and measure the distance between space coordinates. In this example the coordinates (pointed by the arrows) are 8 and 9.6. Thus the length of the rod is 1.6 meters in the frame F. This is called length contraction and this is why we say moving trains are shorter.

The main point to keep in mind are that in special relativity, an event in spacetime might not have same set of coordinates in different frames of reference. To map an event from one coordinate chart to another we must always use Lorentz transformation. If this is followed honestly, then all the paradoxes of special relativity can be resolved. One such paradox is “twin paradox” which I will explain in next post.

If you are wondering how I made these cool Minkowski graphs then head over here.

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June 18, 2015

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The holographic view

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A modern viewpoint on special relativity (I)

In this series of blog posts, I will explain our current understanding of spacetime using the notions of relativity.

Special relativity originates from two simple principles but its predictions challenge our day to day experiences. For instance, according to the relativity, a person in the train station will see the the length of a moving train a little smaller than that of stationary train (of same type). Furthermore, the same person will notice that the clocks hanging in the train ticking  little slower than that on her wrist. These phenomena are termed as “length contraction” and “time dilation”. The magnitude of these contractions and dilations depends directly on the relative velocity of the train and the person standing on the platform. Now, commonly, the relative speed of train is so low that one doesn’t see any appreciable changes. Cosmic particles which travel at very high speeds are known to have longer half lives than their laboratory counterparts. But convincing the audience that relativity is a correct theory of the nature is not the purpose of this post. Assuming that relativity is a valid theory, the aim of this post is to describe and discuss the theory.

In my opinion, special relativity is all about the properties of spacetime in fairly small region and how various observers record the events occurring in that spacetime. The words in italics have precise definitions and meanings in physics so let us spend some time in understanding them. When you read the word spacetime you might think of it as a three dimensional space with time t as separate entity running in the background. This is totally a wrong picture of the spacetime. If you are a physics major, and have had a course in relativity, you might think of it as a strange mixed space of \mathbb{R}^3 and t with 4 coordinates in which the invariant path length is given ds^2 = dx^2+dy^2+dz^2-c^2dt^2, where c is the speed of light.

The problem with the last definition is that it does’t capture the essence of spacetime as such. It is like explaining an alien that the person in the red shirt is a boy. Just like the word “boy” is assigned to a specific creature having certain characteristics and is independent of the shirt he wears, the word “spacetime” is assigned to a specific physical entity and is independent of the coordinates assigned to it by the observer (yes, an observer is someone who assigns the coordinates such that her coordinates are fixed  in the coordinate system). Please note that this is not a standard definition of the observer but it is quite helpful in explaining the subject. In relativity, spacetime is a special set, with many interesting properties, which mathematicians call “Manifold”. And the coordinate system is a map from the spacetime manifold (say the set \mathcal{M}) to the set \mathbb{R}^4. The coordinate chart of a N dimensional manifold is shown in the figure below (courtesy Carroll’s notes)

Manifold

Thus, when an observer is assigning coordinates x, y, z and t to an event, he/she is basically mapping the events in \mathcal{M} to a \mathbb{R}^4 space. When an observer assigns a coordinate system, he/she defines a frame of reference. So when we say “the frame of reference of the observer A”, we mean the coordinates assigned by A.

In physics we postulate that all the physical phenomena, comprising of events, happen in this spacetime manifold. In order to quantitatively analyze them, we require an observer who can assign the coordinates  to these events. First postulate of relativity says that the physics doesn’t depend on the coordinate systems of the observers moving with constant relative velocities with respect to each other. We call all such observers as “inertial observers”. It simply means that if we do the physics calculations using the coordinates of any inertial observer we will find the same results. This shouldn’t be hard to accept as the real physical phenomena take place in the spacetime manifold and no matter which map (coordinate system) we use for quantitative analysis, we must find the same physical results. Now be careful! this postulate talks about the maps which are inertial. So we have to be careful when working with accelerated bodies.

Vectors are mathematical  objects (maps, to be precise) which, by definition, remain unchanged under the coordinate transformations. In relativity, we have 4 dimensional vectors living in the spacetime manifold which remain same in all the inertial coordinate systems. We call such objects as invariants. In fact there is a class of invariants called “Tensors” and we assign all sorts of physical objects (energy-momentum, electromagnetic field etc) to the tensors as they, too, don’t depend on the frame of reference. Now events, which are also physical objects, are assigned to 4 vectors. The distance between two events is called the “path length” and is a very good example of an invariant (actually it is a scalar which is a tensor of rank 0). The second postulate gives us a prescription to make scalars, in terms of what is known as metric. According to this prescription the squared path length is mapped to dx^2+dy^2+dz^2-c^2dt^2 in a coordinate system constructed from x, y, z  and ct . Here c is a constant introduced to match the dimensions of the length squared when multiplied by dt^2 . Simple analysis shows that it must have the dimension of \frac{\text{length}}{\text{time}}. In another coordinate system with the coordinates x^\prime, y^\prime, z^\prime and ct^\prime the squared path length is mapped to dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}-c^2dt^{\prime 2}. Note the same constant c has been used to maintain equality of the frames. Now, due to the invariance of path length, dx^2+dy^2+dz^2-c^2dt^2=dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}-c^2dt^{\prime 2}. This is the starting point of the derivation of the Lorentz transformation in the standard relativity textbooks (although they arrive at this expression using light rays which is misleading sometimes). The Lorentz transformation is a relation between the coordinate systems of two inertial observers, assigned to the same events in the spacetime manifold \mathcal{M}.

In the end of this post I would like to give the celebrated Lorentz transformations. Consider two observers moving with a constant relative speed along one particular direction (say along x axis), then the relation between the coordinates of the observers is (image courtesy Wikepedia)

          Lorentz_transforms_2.svg-2

Here \gamma = 1/\sqrt{1-\frac{v^2}{c^2}}. When physicists did the experiments they found that the value of the constant c is nearly 3\times 10^8meters per second which is the speed of light. The derivation of these transformations can be found in any standard textbook on relativity or in Wikipedia (https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations).