### Space and Time in Special Relativity

Some days ago a non-physicist friend asked me by chat if I could give him a short explanation on the views modern physics has on space. I sent him an email with brief summaries of the concept of space in special relativity, in general relativity, and in quantum gravity speculations. After clarifying some of the doubts the email had left him with, I decided that all that was and should have been blogging material. I just had to translate it to English and expand it, and this is what I shall do now, starting with the Special Relativity part.

As I have been receiving very little feedback recently, I would like to ask my readers to answer, after reading this, if a) those who are not physicists found it understandable, and b) those who are physicists found it a passably good explanation of the issues.

So, let’s get started. The most important new feature about space that arises in Special Relativity (as opposed to Newtonian physics) is that space and time are no longer separated entities but part of a single entity, the much-heard about "space-time continuum". [Trivia question of which I do not know the answer: Does the use of this phrase in popular geeky culture, specially in phrases such as "to disrupt the space-time continuum", trace back to Doc's use of it in Back to the Future, or did it exist previously to the movie, perhaps coming from Star Trek?] But what does it mean exactly to say that space and time are joined in "space-time"? Many people think that it means only that time is the fourth dimension, that objects move through time as well as through space, and that to locate precisely an event one needs to give the full "space-time coordinates", that is four numbers three of which specify position in space and the fourth position in time. All of this is true but not very exciting, it is not the distinctive feature of Einstein's Special Relativity (it is true also in Newtonian physics) and it is not the reason why it is worth talking of "spacetime" as a single entity. The reason for this is rather that time can be "mixed" with spatial dimensions in the same way spatial dimensions can be mixed among themselves.

Imagine you are standing, looking forward with one arm pointing to each side. Your position defines a front-back direction, and a left-right direction which is orthogonal to the other one. ("Orthogonal" is math jargon for "perpendicular" -a strange case in which the jargon word is shorter and easier to say than the common one!) The distinction between these two directions is clear and real -but is relative to your point of view. For another person who is standing like you but rotated in some angle with respect to you, the front-back direction will be different from yours. For example if he is rotated a small angle to the right, then his "front" direction is a combination of your "front" direction and your "right" direction.

In just the same way, observers who are in different states of motion have different "time" and "space" directions. Suppose now you are standing like before, and the other person is not rotated but in motion, moving forwards with respect to you. Your standing defines in the 4-dimensional world a "time" direction and a "front" spatial direction (as well as the other two space directions which do not concern us now). For the other person, there is also a "time" direction and a "front" spatial direction, but these do not correspond to yours. His "time" direction is a combination of your "time" and your "front" directions, in much the same sense as the rotated person's "front" is a combination of your "front" and your "right" directions.

There are more analogies between both cases. If you and a friend standing just by your side but rotated with respect to you calculate the distance to something, you will find the same answer even though you "split" it differently among the different directions. Perhaps what is 5 meters forwards for him will be pehaps 3 meters to the right and 4 meters forwards to you, but the "real distance" (5 meters in this case) is the same for both of you. Pythagoras' Theorem ensures that if

As long as they are not moving, that is. Let's go back to the case when your friend is moving forwards with respect to you. Suppose you and your friend have both rulers and clocks, to measure both spatial and temporal distances between events. Then between any pair of events, you will measure a certain spatial distance

It is easy to see from this how other celebrated fact about relativity emerges, the fact that the speed of light is an absolute. The formula above for

There is one last question my friend made to me, and which I have not answered so far. OK, there is a difference between time and space in that time has a – in a formula where the three spatial dimensions have a +. How does this relate to the ordinary distinction between space and time, to facts such as “we can go backwards in space but not in time”? What other differences between space and time follow from this sign difference? Do ALL of them come from it?

I think that the answer to the last question is yes: all differences between time and space must ultimately be traced to this sign difference. But that does not mean I can spell out a complete answer for the other two questions, partly because the conceptual status of our ordinary notion of “flow of time” is not very clear; it is something related to philosophy and psychology as well as physics. But some of the differences between time and space can be cleared by physics and made more intuitive than just a minus sign. A following post will explain these issues. After that, we’ll be ready to get into General Relativity.

As I have been receiving very little feedback recently, I would like to ask my readers to answer, after reading this, if a) those who are not physicists found it understandable, and b) those who are physicists found it a passably good explanation of the issues.

So, let’s get started. The most important new feature about space that arises in Special Relativity (as opposed to Newtonian physics) is that space and time are no longer separated entities but part of a single entity, the much-heard about "space-time continuum". [Trivia question of which I do not know the answer: Does the use of this phrase in popular geeky culture, specially in phrases such as "to disrupt the space-time continuum", trace back to Doc's use of it in Back to the Future, or did it exist previously to the movie, perhaps coming from Star Trek?] But what does it mean exactly to say that space and time are joined in "space-time"? Many people think that it means only that time is the fourth dimension, that objects move through time as well as through space, and that to locate precisely an event one needs to give the full "space-time coordinates", that is four numbers three of which specify position in space and the fourth position in time. All of this is true but not very exciting, it is not the distinctive feature of Einstein's Special Relativity (it is true also in Newtonian physics) and it is not the reason why it is worth talking of "spacetime" as a single entity. The reason for this is rather that time can be "mixed" with spatial dimensions in the same way spatial dimensions can be mixed among themselves.

Imagine you are standing, looking forward with one arm pointing to each side. Your position defines a front-back direction, and a left-right direction which is orthogonal to the other one. ("Orthogonal" is math jargon for "perpendicular" -a strange case in which the jargon word is shorter and easier to say than the common one!) The distinction between these two directions is clear and real -but is relative to your point of view. For another person who is standing like you but rotated in some angle with respect to you, the front-back direction will be different from yours. For example if he is rotated a small angle to the right, then his "front" direction is a combination of your "front" direction and your "right" direction.

In just the same way, observers who are in different states of motion have different "time" and "space" directions. Suppose now you are standing like before, and the other person is not rotated but in motion, moving forwards with respect to you. Your standing defines in the 4-dimensional world a "time" direction and a "front" spatial direction (as well as the other two space directions which do not concern us now). For the other person, there is also a "time" direction and a "front" spatial direction, but these do not correspond to yours. His "time" direction is a combination of your "time" and your "front" directions, in much the same sense as the rotated person's "front" is a combination of your "front" and your "right" directions.

There are more analogies between both cases. If you and a friend standing just by your side but rotated with respect to you calculate the distance to something, you will find the same answer even though you "split" it differently among the different directions. Perhaps what is 5 meters forwards for him will be pehaps 3 meters to the right and 4 meters forwards to you, but the "real distance" (5 meters in this case) is the same for both of you. Pythagoras' Theorem ensures that if

**dx**and**dy**mean distance forwards and distance sideways for you,**dx'**and**dy’**mean distance forwards and sideways according to him, and**ds**is the "true distance", then it will be true that**dx^2 + dy^2 = dx’^2 + dy’^2 = ds^2.**(Make a drawing if you are not convinced. In the particular example before**dy'**was 0, but it need not be. Of course, "^2" means "squared".) You and your friend see a different forward distance and a different sideway distance to the point you are looking at, but the total distance**ds**comes out the same. If we allow your friend to be also tilted in the vertical direction, and we look at points that are not in the same plane as yours, the formula generalizes to Pythagoras' theorem in three dimensions,**dx^2 + dy^2 + dz^2 = dx’^2 + dy'^2 + dz'^2= ds^2**, where**dz**is the vertical distance according to you, and**dz’**the vertical distance according to him. The important thing is that there is an absolute "spatial distance" which can be split differently in a forward part, a sideways part and a vertical part for different observers, but that comes out the same for every observer.As long as they are not moving, that is. Let's go back to the case when your friend is moving forwards with respect to you. Suppose you and your friend have both rulers and clocks, to measure both spatial and temporal distances between events. Then between any pair of events, you will measure a certain spatial distance

**ds**(which comes from the three separate spatial distances in the three directions as we saw) and a temporal interval**dt**. Your friend will measure a*different*spatial distance**ds**, and a*different*temporal interval**dt’**. But both of your measurements will add up to the same "absolute spacetime distance"**DS**, defined by the formula:**ds^2 – dt^2 = ds’^2 – dt’^2 = DS^2**. The only difference with the usual Pythagorean formula is a – replacing a +. (After all,*some*difference exists between time and space!) But what matters is that, just as "forward distance between points" and "sideways distance between points" are not absolute concepts but depend on which way the observer is looking (but the "spatial distance between points" does not depend on that), it turns out that "spatial distance between events" and "temporal interval between events" are not absolute but depend on how the observer is moving (but the "absolute spacetime distance" does not depend on that). This mixture of time and space is the defining characteristic of the Special Theory of Relativity.It is easy to see from this how other celebrated fact about relativity emerges, the fact that the speed of light is an absolute. The formula above for

**DS^2**subtracts a time squared from a distance squared; this would not be possible if we were not measuring time and space in the same units. To use the same units for space and time, there must be a velocity which is absolute and serves as conversion factor. This velocity is the speed of light. The formula for**DS^2**is correct if we measure for example time in seconds, and distance in light-seconds, defined as the distance traveled by light in a second. (If we were measuring time and space in the usual units, like meters and seconds, the formula would be**DS^2 = ds^2 – c^2 dt^2**, the appearance in it of the speed of light**c**showing that it is an universal constant of nature.)There is one last question my friend made to me, and which I have not answered so far. OK, there is a difference between time and space in that time has a – in a formula where the three spatial dimensions have a +. How does this relate to the ordinary distinction between space and time, to facts such as “we can go backwards in space but not in time”? What other differences between space and time follow from this sign difference? Do ALL of them come from it?

I think that the answer to the last question is yes: all differences between time and space must ultimately be traced to this sign difference. But that does not mean I can spell out a complete answer for the other two questions, partly because the conceptual status of our ordinary notion of “flow of time” is not very clear; it is something related to philosophy and psychology as well as physics. But some of the differences between time and space can be cleared by physics and made more intuitive than just a minus sign. A following post will explain these issues. After that, we’ll be ready to get into General Relativity.

## 8 Comments:

Hola Ale!!!

Soy tu non-physicist friend?

Estoy leyendo tu Blog, dame tiempo, pero me parece muy buena la idea de explicar algunas cosas para non-physicist people, te ahorrarías esos extensos y completos mails que te hago escribir, JeJe...

Abrazo,

Iván

By Iván, at 4:35 PM, February 18, 2006

Gracias por comentar Ivo! No, en este caso no me estaba refiriendo a vos (sino a otro que me pidio unas explicaciones hace pocos dias) pero me alegra que te parezca buena la iniciativa. No dudes en comentar preguntando si hay cosas que no estan claras o con sugerencias para otras £explicaciones"...

By Alejandro, at 6:51 PM, February 18, 2006

La £ debio ser ". Un caso de la mano apretando publish mientras el cerebro todavia esta procesando el error...

By Alejandro, at 6:52 PM, February 18, 2006

I think I may have understood... something.

Seriously, I think it's a good job of explaining something clearly not simple.

By Euge, at 9:35 AM, February 19, 2006

I too found it a good explanation (thanks).

Do feel encouraged to do more posts like this one.

By Anonymous, at 5:44 PM, May 01, 2006

Rob MacRiner rmacriner@sympatico.ca , robmacriner@hotmail.com

Answer to Question: Why does time seem to exist only in a forward direction?

Time seems to only exist in a forward direction because the universe is expanding. If the Universe reaches Critical Velocity and starts to contract ....then time, as we measure time will reverse according to the Big Bang / Big Crunch Theory. The reason for this is that time does not exist without change or movement..... (change or movement of particle matter or energy as we know it). If matter has no movement either expanding or contracting then time does not exit for that matter. However Time can exist around non moving particle matter if something is either expanding or contracting around it.

If the expansion of matter increases as in the case of our universe, or an expanding object, or even light...then time increases relative to the rate of expansion. Example: if carbon A is heated and expands faster than carbon B (which is not heated) then time increases in carbon A relative to carbon B...However as Einstein pointed out...time is relative to the observer...and you need something of contrast to make that comparison....fortunately our universe offers lots of contrast ...otherwise we would have a very difficult time figuring this out. Time being relative to the observer can exist at different speeds based on the rate of expanding matter. If you are on riding on a beam of light than time is much different than your friend riding on a sound wave.Of course time is relative to the observer, therefore your time is much faster only to him, or any body else who is not on a beam of light.

If matter contracts or condenses then time actually reverses…as in the case of a contracting universe…so Planks Quantum would be measured as zero time for the entire Universe…and time starts at the point of the Big Bang (once matter is on the move again)… In the case of a black hole, relative to our expanding universe)... there is also no time. (except for matter being sucked into a black hole….this matter would be reversing in time, until at which point it becomes part of the black hole mass, then time (in a Black Hole) as in Planks Quantum is zero….which is odd because the Universe is still expanding around the black hole…but it is consistent with the theory that. Time can exist around “non moving matter” if something is either expanding or contracting

Time as we know it is measured in a forward direction and will continue until the point of critical velocity…at which point time starts to reverse…and for a brief moment…the point where the Universe changes from expanding to contracting…time will again be zero…as in Planks Quantum. However…during the forward direction of time…(while the Universe is expanding)…black holes are continuing to suck up matter…and should in theory at some point converge with other black holes….Therefore…as the universe is expanding from the big bang…there is multitude of matter which is not expanding (black holes)…which might well be unexploded Planks Quantum matter from the big bang…and the black holes with their massive gravitational force are sucking up matter which was attempting to expand but was not able to overcome the stronger force of the black hole…like mini-Plank Quantum’s converging within the universe …When the Universe reaches Critical Velocity and then all matter in our Universe starts to contract…heading towards the Big Crunch….the multitude of black holes converging (up to that point) should in theory rapidly increase the speed of reverse time …acting as an accelerant force of a contracting Universe with there collective gravitational force …so the reverse of time.(the journey the contracting Universe is taking towards the Big Crunch)...should happen much quicker than the time it took for the Universe to go from the Big Bang to Critical Velocity…That is of course Time relative from the Big Bang to Critical Velocity ……in contrast to …….Time Relative from Critical Velocity to the Big Crunch..… Rob MacRiner rmacriner@sympatico.ca Nov 2007

By Robert, at 3:14 PM, November 13, 2007

So, I do not actually believe it is likely to work.

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By Allen, at 10:38 AM, November 28, 2011

Ain't no such thing as time (hours, minutes, weeks, years, decades, centuries, millennia, etc.) Time is simply a human measure of the condition of continuation of existence. Physicists are making the universe a lot more complicated than it actually is. Time? Bah, Humbug!!!

By Anonymous, at 4:05 PM, December 17, 2012

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