Manifolds Unfolded

We inhabit a world defined by three spatial dimensions—length, breadth, and height. So when someone claims to perceive a fourth dimension, our natural response is skepticism. After all, how can something so fundamental to our experience be extended? Yet, what if the fourth dimension isn’t just a speculative fantasy but a conceptual necessity? What if understanding it requires not more eyesight, but more insight—an unconventional, yet logical reimagining of how reality is structured?

Time Is A Fourth dimension

Yes, you are right. Time can be regarded as a fourth dimension. Let’s begin understanding with a simple example. Suppose, you have a meeting with your friend at a coffee shop located on the 1st floor at the corner of 5th Avenue and 50th Street in New York City. So, even if you and your friend both know the exact location in space for the place of meeting, the meeting is not possible unless the two of you meet there at a particular time. Of course, if you visit the place at 9 AM in the morning of July 28, 2024 and your friend, on the other hand, visits the same place at 9 PM of July 29, 2024; then the meeting won’t be possible. So, it is essential to agree upon a specific point of time for the ‘meeting event’ to occur. Thus, it is not sufficient to know only the 3 co-ordinates of space but also one coordinate of time for the meeting to take place. So, it is not enough to specify the space coordinates (Street, Avenue, Floor) which is (50, 5, 1). We do need to specify a fourth co-ordinate which is the time of the meeting i.e. 9 AM of July 28, 2024. Thus, we have to specify a 4-tuple (Street, Avenue, Floor; Time) given by (50, 5, 1; 9 AM, 28 July, 2024) to uniquely determine the event ‘You meeting with your friend’.

We can thus consider a 4-dimensional space-time world with 3 co-ordinates of space and one co-ordinate of time. Any point in this 4-dimensional space is called an event and is given by a 4-tuple (x, y, z; t). The first 3 specify the ‘where’ of the event, and the last one specifies the ‘when’ of the event.

Each Physical Object has 4 Dimensions

Consider your house which is located at the point (x, y, z) in 3-space. Then what meaning does the fourth coordinate, the time coordinate carries? Let me explain. If I say your office has the co-ordinates (x, y, z; t) in the 4-dimensional space-time, we mean that your office is located in the 3-space at (x, y, z) and your office is time t old, starting from the initial time reference point (say when your house was inaugurated). Then as time would progress, the first 3 space coordinates will remain the same for your house, but the fourth time coordinate will keep on increasing and if we add a slider to the time axis, then dragging the slider forward would gradually take you to the future state of the house, getting older and older and eventually demolished by the ravages of time, unfortunately.

I hope this example makes clear the meaning of the time dimension of a physical object or event.

Space Dimension vs. Time Dimension: How to Compare

Clearly, we can sense that time dimension is not quite the same as the 3 space dimensions. For instance, we can go back and forth in space, but in time, we are allowed to move only in the forward direction, from past to future. Space and Time are different kinds of physical quantities and are measured in different ways. For example, time intervals are measured by a clock, for which tick-tock sound indicates passing of seconds and ding-dong indicates passing of hours. On the contrary, space intervals / distances are measured by yardsticks. So, clearly, there seems to be no way to compare space intervals with time intervals. After all, we cannot turn a yardstick into a clock or vice-versa. So apparently, time and space are incomparable physical quantities and have different units. Thus, it looks like there is no means to convert a time measurement into equivalent space measurement and vice versa. But what if I tell you, there is a way we will be discussing shortly – the space-time equivalence.

Before that let us get some intuition of how we should think about or visualize the 4 dimensional space-time.

Visualizing the Time Dimension

Despite such contrasting features between time dimension and space dimension, we can view time as a fourth dimension in the realm of physical events. In choosing time as the fourth dimension, we shall find it much simpler to visualize the 4 dimensional picture. By, now you should be confident enough to regard yourself as a 4 dimensional entity in space-time. Below is a schematic depiction of the space-time geometry showing you as a 4-dimensional entity in space-time.

Space-Time Equivalence: The Genius of Einstein

The three dimensions of space viz. length, breadth and height, all have the same unit of length – Inches, Centimeters, Meters etc. However, Time is a different physical quantity measured in units of time – seconds, minutes, hours etc. Thus a natural question that should bother everyone at this point is : How do time and space compare?

It turns out that there is a reasonable way of comparing space and time! Let’s try to motivate this from our day-to-day vocabulary. We often say that someone lives within 20 minutes of downtown by bus or five hours by train. Notice that, we are referring to a distance, which is a space measurement, but we are using time measurement to specify the distance 20 minutes by bus or five hours by train! If we reflect upon it for a moment, we realize that this response stems from our experience or knowledge about the speed of the bus / train. If we know that the bus travels at a speed of 50 kilometers / hour. Then, we can calculate the distance as ( 50/60 x 20 ) kilometers i.e. about 16. 667 kilometers. Similarly, if the train travels at a speed of 200 kilometers / hour, then we can calculate the distance as ( 200 x 5 ) kilometers i.e. 1,000 kilometers.

So, it turns out that we can convert time into distance by multiplying by a speed quantity, as a conversion factor because,

Thus, what we need is a standard speed to agree upon, so as to compare time and space. This genius idea of using speed as the conversion factor occurred to the genius Physicist Albert Einstein.

It is quite clear that this conversion factor between space and time must be very much fundamental and general in nature. It should remain invariant regardless of human initiatives or physical circumstances.

Speed – But Speed of What?

We all agree up to the point that we need some sort of very fundamental and invariant speed quantity as a conversion factor between time and space. But, the question is – speed of what?
It turns out that the only speed known in Physics that possesses the desired generality is the speed of light in empty space (i.e. vacuum). We call it simply The Speed of Light. We can also call it The Propagation Speed of Physical Interactions for valid reasons.

Speed of Light: A Crucial Piece of the Puzzle

The first to measure the speed of light was Olaus Roemer in 1676. He observed the eclipses of Jupiter’s moon Io and noted discrepancies in their timing, which he interpreted as light traveling at a finite speed. While he didn’t get the exact value, this was the first attempt to measuring the speed of light. Around 1848-1849, French Physicist Hippolyte Fizeau used a toothed wheel apparatus to perform absolute measurements of the speed of light in air. His measurements were quite close to the actual speed of light which is 299,792,458 meters per second (i.e. almost 3 x 10⁸ meters per second)! We denote this speed by .

We can now decide to compare space and time using this constant as the conversion factor. In the same way we said 5 hours by train, we can now say 6,000 seconds by light to indicate a distance of 6,000 meters i.e. 18 x 10¹¹ meters in space. This means that the amount of time required for light in space to traverse through a distance of 18 x 10¹¹ meters is 6,000 seconds !

So, now we are capable of measuring an interval on the time axis in terms of units of length, allowing us to convert time measurement into space measurement and vice-versa. An interval of time units will correspond to a length or distance of in space units.

Four Dimensional Distance in Space-Time

We have successfully arrived at a 4 dimensional model with 3 space dimensions and 1 time dimensions, termed as the space-time and comprising of all possible events, given by 4-tuples (x, y, z; t). Now think of two events A and B with space-time coordinates and . The 3-dimensional distance between the locations of the two events in 3-space is given by the Euclidean distance between the points , which is:

Which is the length of the straight line path joining the two points .

If the concept of the fourth coordinate has any practical validity, then we should be able to combine the figure

for space separation with the figure

for time separation to obtain a single figure – which can then be regarded as the four dimensional distance between the two events given by the 4-tuples.

What comes immediately to our mind is the Euclidean Distance in 4-dimensions between the points given by . However, we also realize that in doing so, we are losing the distinction between the whole different physical quantities – space and time.

Our intuition says, we should express the distance between two space-time events using some unconventional way of measuring distance, one which would keep the difference of space and time intact.

This issue is resolved by a minor adjustment in the expression:

which is known as Einstein’s 4-dimensional distance between two events in space-time. Invoking this kind of weird and new distance formula (which can spit out real as well as imaginary values) gives rise to a whole new kind of geometry – called the Lorentzian Geometry (a particular case of Pseudo-Riemannian Geometry).

Minkowski’s Alternative in Lorentzian Geometry – Time as Purely Imaginary Coordinate

At first glance, this new notion of distance might invoke an initial discomfort, as the distance proposed above is really strange and seems to be coming out of the blue. However, Einstein’s teacher Hermann Minkowski was wise enough to come up with a remedy to ease this discomfort. He proposed that, we should consider the fourth coordinate, the time coordinate as a purely imaginary quantity – instead of considering the time coordiante as one should consider it as , a purely imaginary quantity. Thus our 4-tuple in space-time should be . Then, invoking the usual distance formula (from Euclidean Geometry), we would get,

, which is same as the Einstein’s 4-dimensional space-time distance given earlier! This distance is referred to as the Lorentzian distance (aka. Minkowski Distance) and it can be real as well as complex, depending upon whether the expression inside the square root is non-negative or negative. If the expression inside the square roots i.e. is positive, we say that the two events have a space-like separation between them, and we use the phrase – spatially separated. On the other hand, if is negative, we say that the separation between the two events is time-like, and use the phrase – the events are temporally separated. To remember the contribution of Minkowski, Lorentzian Geometry is also called Minkowskian Geometry.

Lorentzian Geometry : In Rigorous Terms

A smooth, connected, manifold equipped with a Lorentzian Metric i.e. a smooth metric tensor, which is symmetric and non-degenerate (but not necessarily positive-definite) is called a Lorentzian Manifold. Those who are not familiar with the term smooth metric-tensor can think of it to be a smooth assignment of bilinear forms to the tangent spaces at each point of the manifold .

This is quite similar to the Riemannian Geometry, where we assign an inner product smoothly to the tangent space at each point ( is known as the Riemannian Metric). Note that, Inner products are precisely symmetric, non-degenerate and positive-definite bilinear forms. The only difference is that in case of Lorentzian Metric / Pseudo-Riemannian Metric, we relax the positive-definiteness hypothesis for the metric-tensor .

Symmetry of means : for every . Non-degeneracy means that,

.

The 3 Different kinds of Tangent Vectors

In Lorentzian Geometry, every tangent space is a Pseudo-Euclidean vector space and there are 3 kinds of tangent vectors attached to a point on the manifold – time-like, null and space-like depending upon whether the Lorentzian length of the tangent vector is an imaginary number, zero or a positive real number.

Einstein’s Theory of Space-Time Relativity : Modeled by Lorentzian Geometry

We all are familiar with the name of Albert Einstein‘s exceptionally genius and groundbreaking Theory of General Relativity of Space-Time. The principal premise of this theory is that space-time can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1). Signature refers simply to the fact that there are 3 space dimensions and 1 time dimension, The space-time is locally modeled over the flat Minkowski space , which is just the smooth manifold but now equipped with the flat Minkowski Metric given by .

Riemannian Geometry v/s Lorentzian Geometry

There are similarities as well as contrasts between Riemannian Geometry and Lorentzian Geometry. For example, analog of The Fundamental Theorem of Riemannian Geometry holds in case of Lorentzian / Pseudo-Riemannian Geometry. Thus we can talk about the Levi-Civita connection and Curvature Tensor. Thus allows us to rigorously formulate the concept of space-time curvature.

However, unlike Riemannian Geometry, not every smooth manifold admits a Pseudo-Riemannian Metric of a given signature, due to topological obstructions. Also, a submanifold does not always inherit the structure of a Lorentzian Manifold, which is a sharp contrast to the corresponding Riemannian case.

In Riemannian Geometry, a compact Riemannian manifold is complete by the Hopf-Rinow Theorem. However, the analogous result is not true for Lorentzian Geometry – there exist Pseudo-Riemannian manifold that are compact but not complete!

Final Thoughts

This new kind of 4-dimensional geometry of space-time plays the central role in Einstein’s Theory of Relativity. It explains the distortions in space-time fabric caused by the presence of massive objects like stars and planets. This theory explains Gravity as a consequence of curvature in space-time.

As opposed to Newton, this new theory developed by Einstein views time as a relative quantity, not as something absolute – explaining seemingly counter-intuitive phenomena such as time dilation in outer space.

In my next blog post, We will delve deep into the Theory of General Relativity and understand space-time curvature, time-dilation and gravity in light of this ingenious theory.