What is an intuitive explanation of the Lorentz transformation?
Lorentz transformations are only important because of Lorentz invariance. So I will explain both. I think it's easiest to understand Lorentz transformations and Lorentz invariance if you start by thinking about three-dimensional rotations and rotational invariance.
The concept of rotation is very familiar to all of us because of the curious fact that all objects can be rotated. If you stop and think about it for a moment, this is a very curious fact. Not all objects can be stretched, for example: a rubber band can be stretched, but if you try to stretch a noodle more than a tiny bit, it'll simply break. Without exception, all objects can be rotated, in the sense that they can exist in any orientation; so it's worthwhile to consider why this is. What makes rotation special in a way that stretching is not?
The answer is that the laws of physics that govern our universe exhibit rotational invariance, which is the technical way of saying that the rotated version of an isolated system exhibits the same behaviour as the unrotated version, that isolated systems cannot determine their orientation, or that the laws of physics do not "discriminate" against orientation. For example,
A key fact about rotations is that they leave all distances unchanged. This guarantees, for example, that an object's diameter is not changed when it is rotated. It also guarantees that angles are not distorted. This is the defining property of rotations.
Now, a rotation is a type of Lorentz transformation. But rotations do not go far enough. For it is known that not only can all objects be rotated, but also that all objects can be boosted. For a moment I'm going to pretend that the universe is Newtonian, so I can explain what a boost is. A boost changes the velocity of a system uniformly (like how a translation changes the position uniformly and a rotation changes the orientation uniformly). A boost of 10 m/s to the right, for example, would convert:
Note also that a boost is not the process of accelerating a system from its old velocity to a new velocity. A boost is a function that maps the entire history of a system to a different history, in which all the velocities are shifted by the same amount.
Boosts appear to be "protected" by a symmetry similar to rotational symmetry, in which the laws of physics all treat a boosted system the same as the original system. Hence an isolated system can't know its velocity just as it can't know its orientation. [3] Here are some examples of physical consequences of this fact:
Now this picture so far was the picture in Newtonian physics. The problem is that this picture is not completely accurate. While it is true that a system can't know its own velocity, that there is a symmetry regarding velocity, it doesn't work exactly the way I said. There is light, whose speed cannot be changed; it always travels at the speed of light. The speed of light is also a universal speed limit. Boosting a system does affect all parts of the system uniformly, but not in the naive Newtonian fashion. If in a stationary box there is an electron travelling at .5c to the right, then the box, when boosted so that it's travelling at .5c to the right, does not contain an electron travelling at c to the right, but rather one travelling at .8c to the right.
These strange properties can be explained by the fact that a boost, unlike a rotation, doesn't only affect space. It also affects time. A rotation changes the positions of the components of a system, but doesn't change the time point of any event; that is so obvious that we don't even mention it, usually. But a boost, or a combination of a boost and a rotation, can change the time at which an event occurs. It can mix up space and time, just like rotations mix up space. One consequence of this is that moving clocks run slower than stationary clocks; the moving clock, which can be viewed as a boosted version of the stationary clock, moves through space faster than the stationary clock but also moves through time more slowly. The boost mixed up space and time.
The above discussion of rotations and Newtonian boosts should be intuitive enough, but if you really want an explanation of the Lorentz transformation and Lorentz boosts, then I have to break out some algebra. I will try to leverage the intuitive exposition previously given in order to make this as painless as possible.
A rotation may change the position of a particle but it does not change the distance between two particles,
[math]d^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2[/math]
Rotations don't change angles either, nor do they change time. That's what makes rotations what they are. So really, rotational invariance is the statement that if a system is transformed in a way that leaves all its internal distances and angles invariant, as well as all time points, then its behaviour doesn't change; it can't tell the difference, so to speak. The formula above characterizes rotations, and the statement of rotational invariance gives rotations special status among all the possible transformations and reveals a deep truth about the universe.
But rotations are just a special case of a more general kind of transformation; that is the Lorentz transformation, and the laws of physics as we know them are Lorentz invariant. A Lorentz transformation leaves the following space-time interval invariant:
[math]s^2 = (c\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2[/math]
or
[math]s^2 = (c\Delta t)^2 - d^2[/math]
A rotation is a type of Lorentz transformation because a rotation leaves [math]\Delta t[/math] and [math]d^2[/math] separately invariant, so it of course leaves [math]s^2 = (c\Delta t)^2 - d^2[/math] invariant. But we can also see that different kinds of Lorentz transformations are possible! For example, consider the following Lorentz transformation:
[math]t' = \frac{5}{4}t + \frac{3}{4c}x[/math]
[math]x' = \frac{3c}{4}t + \frac{5}{4}x[/math]
[math]y' = y[/math]
[math]z' = z[/math]
That is, if an object is located at the spatial coordinates [math](x, y, z)[/math] at time [math]t[/math], then its Lorentz-transformed image is located at the spatial coordinates [math](x', y', z')[/math] at the time [math]t'[/math], calculated using the above formulae.
If you do the math, you'll see that if all points have their x and t coordinates modified according to this transformation, then for a given pair of points, their coordinates after, [math](t'_1, x'_1, y'_1, z'_1)[/math] and [math](t'_2, x'_2, y'_2, z'_2)[/math] will have the same space-time interval between them as their coordinates before, [math](t_1, x_1, y_1, z_1)[/math] and [math](t_2, x_2, y_2, z_2)[/math].
In the case that [math]t' = t[/math] you have a rotation, but for this particular example that isn't the case, so it isn't a rotation. The following is a rotation:
[math]t' = t[/math]
[math]x' = \frac{3}{5}x + \frac{4}{5}y[/math]
[math]y' = -\frac{4}{5}x + \frac{3}{5}y[/math]
[math]z' = z[/math]
Notice two differences between this rotation and the previously given Lorentz transformation, which is a boost (and we'll see why shortly). The boost mixes up the x axis and the time axis. The rotation mixes up the x and y axes and leaves time alone. Also the rotation has a minus sign, whereas the boost has all plus signs. The reason for this difference is that the space-time interval has a plus sign for time and minus signs for space. So boosts are not exactly like rotations (they're actually "hyperbolic rotations") but my personal intuition for Lorentz transformations does come from imagining them as general rotations which may mix up space and time. That's the intuitive explanation.
To understand the effect of this boost, first imagine an object sitting at the origin. It has [math]x = y = z = 0[/math]. Now apply this transformation. [math]y'[/math] and [math]z'[/math] will of course be zero, but
[math]t' = \frac{5}{4}t + \frac{3}{4c}x = \frac{5}{4}t[/math]
[math]x' = \frac{3c}{4}t + \frac{5}{4}x = \frac{3c}{4}t[/math]
so the equation of the trajectory after the Lorentz transformation is
[math]x' = \frac{3c}{5}t'[/math]
In other words, this Lorentz transformation maps a stationary object onto an object moving along the positive x axis at .6c. So it truly is a boost; it changes a stationary object to a uniformly moving one.
This kind of Lorentz transformation, that moves around points in space and time, is an active Lorentz transformation. But now imagine that this object travelling at .6c to the right is actually another observer, who we'll name Alice. According to Alice, she herself is stationary, since a moving system can't know its own velocity. So according to her, her x, y, and z coordinates are all zero, and she's only moving through time. But since you see Alice moving to the right at .6c, she must see you as moving to the left at .6c even though you perceive yourself to be at the origin. That is, the coordinates that you assign to yourself are [math]x = y = z = 0[/math], but according to Alice---in Alice's frame of reference, your coordinates satisfy [math]x'' = -\frac{3c}{5}t''[/math]. (I'm using the double primes to denote Alice's frame of reference.)
If you take the form of the active Lorentz transformation in coordinates that you used to transform Alice from stationary to moving to the right at .6c, then it maps the coordinates of the trajectory of an object stationary at the origin to the coordinates of the trajectory of an object moving to the right at .6c which implies that its inverse transformation takes the coordinates of the trajectory of an object moving to the right at .6c---Alice's coordinates in your frame of reference---to the coordinates of an object stationary at the origin---Alice's coordinates in her frame of reference. Thus it's not hard to see that the inverse of this active Lorentz transformation can be used to transform the coordinates of a point in one frame of reference (yours) to the coordinates of the same point in another (Alice's). This is called a passive Lorentz transformation.
Whether you choose to consider Lorentz transformations active or passive doesn't change the fact that they're defined to keep the space-time interval invariant. They have the same mathematical properties. But the passive transformations reveal insight into how different observers observe differently, which is what makes relativity unintuitive and revolutionary.
Let's say Bob, facing north, throws a ball 10 metres forward. Alice is facing northeast. To her, the ball only moved [math]5\sqrt{2}[/math] metres forward; it also moved [math]5\sqrt{2}[/math] metres to the left. However, both Alice and Bob agree that the ball moved 10 metres in total. To Bob, the distance is [math]\sqrt{10^2 + 0^2 + 0^2}[/math] metres. To Alice, the distance is [math]\sqrt{(5\sqrt{2})^2 + (5\sqrt{2})^2 + 0^2}[/math] metres. Of course they agree; rotations preserve distances, so a rotated observer always sees the same distance as the original. But they disagree on the individual components. This is nothing strange; it's totally intuitive since we're all familiar with how rotations work.
But if rotations can mix up spatial coordinates and boosts can mix up space with time, then by that very analogy, a boosted observer might observe a different amount of time between two events compared to the original observer. This is indeed the case. Let's say Bob at the origin times 4 seconds on his wristwatch... at the beginning, his coordinates are [math](0, 0, 0, 0)[/math] and at the end they are [math](4, 0, 0, 0)[/math] where time is measured in seconds and space is measured in light-seconds. Alice on the other hand sees different coordinates for Bob, given by the inverse of the previously given Lorentz transformation:
[math]t' = \frac{5}{4}t - \frac{3}{4c}x[/math]
[math]x' = -\frac{3c}{4}t + \frac{5}{4}x[/math]
[math]y' = y[/math]
[math]z' = z[/math]
To Alice, Bob's initial coordinates are [math](0, 0, 0, 0)[/math], but his final coordinates are [math](5, -3, 0, 0)[/math]. So according to Alice, five seconds have passed and Bob has moved 3 light-seconds to the left, while according to Bob, four seconds have passed and he remained stationary. But both Alice and Bob agree on the space-time interval that Bob traversed, since a Lorentz transformation leaves that invariant.
So you can always boost a clock, but when you do, it will appear to run more slowly. This seems really weird unless you keep in mind that a boost mixes up space and time. When you keep that in mind, a clock appearing to run more slowly when boosted is no different from the way in which someone running at an angle to your line of sight seems to be moving forward more slowly to you than they perceive themselves to. It's just a matter of the different observers using different coordinates.
[1] I've used the word "isolated" because it's important. The relative orientation of objects that are not isolated does have physical consequences. If you only rotate one of two magnets, for example, you can change the distance between their poles, which changes the force and potential energy.
[2] Note that translations also have this property.
[3] If a system knows its own velocity, then it isn't isolated. For example, it may be measuring its speed relative to the air around it, or it may be triangulating its position using GPS signals.
The concept of rotation is very familiar to all of us because of the curious fact that all objects can be rotated. If you stop and think about it for a moment, this is a very curious fact. Not all objects can be stretched, for example: a rubber band can be stretched, but if you try to stretch a noodle more than a tiny bit, it'll simply break. Without exception, all objects can be rotated, in the sense that they can exist in any orientation; so it's worthwhile to consider why this is. What makes rotation special in a way that stretching is not?
The answer is that the laws of physics that govern our universe exhibit rotational invariance, which is the technical way of saying that the rotated version of an isolated system exhibits the same behaviour as the unrotated version, that isolated systems cannot determine their orientation, or that the laws of physics do not "discriminate" against orientation. For example,
- Stretching a rubber band increases its potential energy, but rotating a rubber band doesn't change its energy at all.
- Rotating two permanent magnets by the same amount around the same axis does not does not change the amount of magnetic force between them.
- A rotated box of gas still has the same pressure and temperature, and therefore still (approximately) satisfies the ideal gas law just as before.
A key fact about rotations is that they leave all distances unchanged. This guarantees, for example, that an object's diameter is not changed when it is rotated. It also guarantees that angles are not distorted. This is the defining property of rotations.
Now, a rotation is a type of Lorentz transformation. But rotations do not go far enough. For it is known that not only can all objects be rotated, but also that all objects can be boosted. For a moment I'm going to pretend that the universe is Newtonian, so I can explain what a boost is. A boost changes the velocity of a system uniformly (like how a translation changes the position uniformly and a rotation changes the orientation uniformly). A boost of 10 m/s to the right, for example, would convert:
- a stationary train into a train moving at 10 m/s to the right;
- a train moving at 50 m/s to the right to a train moving at 60 m/s to the right;
- a train moving at 50 m/s to the left to a train moving at 40 m/s to the left
- a train moving at 50 m/s to the left with a human inside running at 5 m/s to the right relative to the train, to a train moving at 40 m/s to the left with a human inside still running at 5 m/s to the right relative to the train.
Note also that a boost is not the process of accelerating a system from its old velocity to a new velocity. A boost is a function that maps the entire history of a system to a different history, in which all the velocities are shifted by the same amount.
Boosts appear to be "protected" by a symmetry similar to rotational symmetry, in which the laws of physics all treat a boosted system the same as the original system. Hence an isolated system can't know its velocity just as it can't know its orientation. [3] Here are some examples of physical consequences of this fact:
- a robot programmed to walk up stairs will be equally able to walk up an escalator (no guarantees about how well they handle getting off at the top, though)
- a ball dropped inside an elevator ascending at constant velocity is as bouncy as it would be on the ground
- an object tossed within a train will travel as far relative to the train as it would travel relative to the ground if tossed within the station
- and, on a larger scale, as the Earth changes direction as it journeys around the Sun, you can't feel which direction the Earth is currently travelling in (though you could consult a calendar if you really wanted to know!)
Now this picture so far was the picture in Newtonian physics. The problem is that this picture is not completely accurate. While it is true that a system can't know its own velocity, that there is a symmetry regarding velocity, it doesn't work exactly the way I said. There is light, whose speed cannot be changed; it always travels at the speed of light. The speed of light is also a universal speed limit. Boosting a system does affect all parts of the system uniformly, but not in the naive Newtonian fashion. If in a stationary box there is an electron travelling at .5c to the right, then the box, when boosted so that it's travelling at .5c to the right, does not contain an electron travelling at c to the right, but rather one travelling at .8c to the right.
These strange properties can be explained by the fact that a boost, unlike a rotation, doesn't only affect space. It also affects time. A rotation changes the positions of the components of a system, but doesn't change the time point of any event; that is so obvious that we don't even mention it, usually. But a boost, or a combination of a boost and a rotation, can change the time at which an event occurs. It can mix up space and time, just like rotations mix up space. One consequence of this is that moving clocks run slower than stationary clocks; the moving clock, which can be viewed as a boosted version of the stationary clock, moves through space faster than the stationary clock but also moves through time more slowly. The boost mixed up space and time.
The above discussion of rotations and Newtonian boosts should be intuitive enough, but if you really want an explanation of the Lorentz transformation and Lorentz boosts, then I have to break out some algebra. I will try to leverage the intuitive exposition previously given in order to make this as painless as possible.
A rotation may change the position of a particle but it does not change the distance between two particles,
[math]d^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2[/math]
Rotations don't change angles either, nor do they change time. That's what makes rotations what they are. So really, rotational invariance is the statement that if a system is transformed in a way that leaves all its internal distances and angles invariant, as well as all time points, then its behaviour doesn't change; it can't tell the difference, so to speak. The formula above characterizes rotations, and the statement of rotational invariance gives rotations special status among all the possible transformations and reveals a deep truth about the universe.
But rotations are just a special case of a more general kind of transformation; that is the Lorentz transformation, and the laws of physics as we know them are Lorentz invariant. A Lorentz transformation leaves the following space-time interval invariant:
[math]s^2 = (c\Delta t)^2 - (\Delta x)^2 - (\Delta y)^2 - (\Delta z)^2[/math]
or
[math]s^2 = (c\Delta t)^2 - d^2[/math]
A rotation is a type of Lorentz transformation because a rotation leaves [math]\Delta t[/math] and [math]d^2[/math] separately invariant, so it of course leaves [math]s^2 = (c\Delta t)^2 - d^2[/math] invariant. But we can also see that different kinds of Lorentz transformations are possible! For example, consider the following Lorentz transformation:
[math]t' = \frac{5}{4}t + \frac{3}{4c}x[/math]
[math]x' = \frac{3c}{4}t + \frac{5}{4}x[/math]
[math]y' = y[/math]
[math]z' = z[/math]
That is, if an object is located at the spatial coordinates [math](x, y, z)[/math] at time [math]t[/math], then its Lorentz-transformed image is located at the spatial coordinates [math](x', y', z')[/math] at the time [math]t'[/math], calculated using the above formulae.
If you do the math, you'll see that if all points have their x and t coordinates modified according to this transformation, then for a given pair of points, their coordinates after, [math](t'_1, x'_1, y'_1, z'_1)[/math] and [math](t'_2, x'_2, y'_2, z'_2)[/math] will have the same space-time interval between them as their coordinates before, [math](t_1, x_1, y_1, z_1)[/math] and [math](t_2, x_2, y_2, z_2)[/math].
In the case that [math]t' = t[/math] you have a rotation, but for this particular example that isn't the case, so it isn't a rotation. The following is a rotation:
[math]t' = t[/math]
[math]x' = \frac{3}{5}x + \frac{4}{5}y[/math]
[math]y' = -\frac{4}{5}x + \frac{3}{5}y[/math]
[math]z' = z[/math]
Notice two differences between this rotation and the previously given Lorentz transformation, which is a boost (and we'll see why shortly). The boost mixes up the x axis and the time axis. The rotation mixes up the x and y axes and leaves time alone. Also the rotation has a minus sign, whereas the boost has all plus signs. The reason for this difference is that the space-time interval has a plus sign for time and minus signs for space. So boosts are not exactly like rotations (they're actually "hyperbolic rotations") but my personal intuition for Lorentz transformations does come from imagining them as general rotations which may mix up space and time. That's the intuitive explanation.
To understand the effect of this boost, first imagine an object sitting at the origin. It has [math]x = y = z = 0[/math]. Now apply this transformation. [math]y'[/math] and [math]z'[/math] will of course be zero, but
[math]t' = \frac{5}{4}t + \frac{3}{4c}x = \frac{5}{4}t[/math]
[math]x' = \frac{3c}{4}t + \frac{5}{4}x = \frac{3c}{4}t[/math]
so the equation of the trajectory after the Lorentz transformation is
[math]x' = \frac{3c}{5}t'[/math]
In other words, this Lorentz transformation maps a stationary object onto an object moving along the positive x axis at .6c. So it truly is a boost; it changes a stationary object to a uniformly moving one.
This kind of Lorentz transformation, that moves around points in space and time, is an active Lorentz transformation. But now imagine that this object travelling at .6c to the right is actually another observer, who we'll name Alice. According to Alice, she herself is stationary, since a moving system can't know its own velocity. So according to her, her x, y, and z coordinates are all zero, and she's only moving through time. But since you see Alice moving to the right at .6c, she must see you as moving to the left at .6c even though you perceive yourself to be at the origin. That is, the coordinates that you assign to yourself are [math]x = y = z = 0[/math], but according to Alice---in Alice's frame of reference, your coordinates satisfy [math]x'' = -\frac{3c}{5}t''[/math]. (I'm using the double primes to denote Alice's frame of reference.)
If you take the form of the active Lorentz transformation in coordinates that you used to transform Alice from stationary to moving to the right at .6c, then it maps the coordinates of the trajectory of an object stationary at the origin to the coordinates of the trajectory of an object moving to the right at .6c which implies that its inverse transformation takes the coordinates of the trajectory of an object moving to the right at .6c---Alice's coordinates in your frame of reference---to the coordinates of an object stationary at the origin---Alice's coordinates in her frame of reference. Thus it's not hard to see that the inverse of this active Lorentz transformation can be used to transform the coordinates of a point in one frame of reference (yours) to the coordinates of the same point in another (Alice's). This is called a passive Lorentz transformation.
Whether you choose to consider Lorentz transformations active or passive doesn't change the fact that they're defined to keep the space-time interval invariant. They have the same mathematical properties. But the passive transformations reveal insight into how different observers observe differently, which is what makes relativity unintuitive and revolutionary.
Let's say Bob, facing north, throws a ball 10 metres forward. Alice is facing northeast. To her, the ball only moved [math]5\sqrt{2}[/math] metres forward; it also moved [math]5\sqrt{2}[/math] metres to the left. However, both Alice and Bob agree that the ball moved 10 metres in total. To Bob, the distance is [math]\sqrt{10^2 + 0^2 + 0^2}[/math] metres. To Alice, the distance is [math]\sqrt{(5\sqrt{2})^2 + (5\sqrt{2})^2 + 0^2}[/math] metres. Of course they agree; rotations preserve distances, so a rotated observer always sees the same distance as the original. But they disagree on the individual components. This is nothing strange; it's totally intuitive since we're all familiar with how rotations work.
But if rotations can mix up spatial coordinates and boosts can mix up space with time, then by that very analogy, a boosted observer might observe a different amount of time between two events compared to the original observer. This is indeed the case. Let's say Bob at the origin times 4 seconds on his wristwatch... at the beginning, his coordinates are [math](0, 0, 0, 0)[/math] and at the end they are [math](4, 0, 0, 0)[/math] where time is measured in seconds and space is measured in light-seconds. Alice on the other hand sees different coordinates for Bob, given by the inverse of the previously given Lorentz transformation:
[math]t' = \frac{5}{4}t - \frac{3}{4c}x[/math]
[math]x' = -\frac{3c}{4}t + \frac{5}{4}x[/math]
[math]y' = y[/math]
[math]z' = z[/math]
To Alice, Bob's initial coordinates are [math](0, 0, 0, 0)[/math], but his final coordinates are [math](5, -3, 0, 0)[/math]. So according to Alice, five seconds have passed and Bob has moved 3 light-seconds to the left, while according to Bob, four seconds have passed and he remained stationary. But both Alice and Bob agree on the space-time interval that Bob traversed, since a Lorentz transformation leaves that invariant.
So you can always boost a clock, but when you do, it will appear to run more slowly. This seems really weird unless you keep in mind that a boost mixes up space and time. When you keep that in mind, a clock appearing to run more slowly when boosted is no different from the way in which someone running at an angle to your line of sight seems to be moving forward more slowly to you than they perceive themselves to. It's just a matter of the different observers using different coordinates.
[1] I've used the word "isolated" because it's important. The relative orientation of objects that are not isolated does have physical consequences. If you only rotate one of two magnets, for example, you can change the distance between their poles, which changes the force and potential energy.
[2] Note that translations also have this property.
[3] If a system knows its own velocity, then it isn't isolated. For example, it may be measuring its speed relative to the air around it, or it may be triangulating its position using GPS signals.