Unraveling The Minkowski Metric: Intuitive Explanation

[aaaa202]

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively.

The quantity:

dx²+dy²+dy²-c²dt²
is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I understand intuitively why the above quantity is invariant. Can someone explain to me?
For me it should rather be something that expresses that light travels the same distance in every same dt, so maybe rather cdt = cdt' but then again dt and dt' are in general different right? So long since I did SR

 

[omega_minus]

You're right t does not equal t' but then again x does not equal x' for velocities along x. The length contraction will make ds smaller while the time dilation makes ds longer by the same amount (gamma). Recall that SR is a 4-D volume preserving theory. If a cube of 1m^3 comes into existence for one second then vanishes again, all observers will agree on how much spacetime was enclosed by it. They will disagree on how long it was spatially (and hence its 3-D volume) and how long it existed, but take the euclidian norm sqrt(x^2+y^2+z^2-(ct)^2) and everyone agrees on how much spacetime was inside the box. In summary: we disagree on how much space there was between two events and we disagree on how much time was between two events, but we ALL agree on how much spacetime was between two events.

 

[Muphrid]

Where's the closest hospital to you? Let's say it's 5 miles away from you. It might be 3 miles north and 4 miles east, and you'd use a distance formula to turn those two components into the full distance. But something you implicitly understand is that it doesn't matter whether you break the distance down into north and east components specifically. The distance is the same regardless of coordinate system.

The distance formula in special relativity is what you have. Even if you rotate or change the coordinate system, the spacetime distance (the "interval" is what we call it so we don't get confused with conventional distance) is the same between two points (events).

 

[pervect]

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively.

The quantity:

dx²+dy²+dy²-c²dt²
is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I understand intuitively why the above quantity is invariant. Can someone explain to me?
For me it should rather be something that expresses that light travels the same distance in every same dt, so maybe rather cdt = cdt' but then again dt and dt' are in general different right? So long since I did SR

If you look at the Lorentz interval between two points on a light beam, it will always be zero, because the distance^2 will equal (ct)^2.

So the constantcy of the speed of light for all observers is equivalent to saying that a Lorentz interval of zero in one frame is zero in all frames.

IT's not intuitively obvious how non-zero Lorentz intervals should transform (it is intuitively obvious via the constancy priciples how zero Lorentz intervals must transform). But I don't believe it has to be " intuitively obvious".

 

[sardar]

I understand your confusion with the Minkowski metric and its invariance in special relativity. The Minkowski metric, also known as the spacetime interval, is a fundamental concept in special relativity that represents the distance between two events in spacetime. It is given by the equation:

ds2 = dx2 + dy2 + dz2 - c2dt2

Where ds is the spacetime interval, dx, dy, and dz are the spatial coordinates, and c is the speed of light. This metric is used to define the geometry of spacetime in special relativity, and it is invariant for all inertial frames of reference.

To understand the invariance of the Minkowski metric, we must first understand the concept of spacetime and how it is affected by the speed of light. In special relativity, space and time are not separate entities, but rather they are combined into a single four-dimensional spacetime. This means that an event in spacetime is described by four coordinates: three spatial coordinates (x, y, z) and one time coordinate (t).

Now, let's consider the motion of an object in spacetime. As the object moves through spacetime, it traces out a path, known as its worldline. This path can be represented by a line in a spacetime diagram, with the spatial coordinates on the x and y axes and the time coordinate on the t axis. In this diagram, the slope of the line represents the object's velocity, and the steeper the slope, the faster the object is moving.

Now, let's imagine two events, A and B, that occur at different points in spacetime. The distance between these two events can be calculated using the Minkowski metric. If the object is at rest, the distance between A and B is simply the spatial distance between them, as time does not play a role. However, when the object is moving, time is affected by its velocity. This means that the time coordinate, represented by the t axis, is stretched or compressed depending on the object's velocity.

The invariance of the Minkowski metric means that the spacetime interval between two events is the same for all observers, regardless of their relative velocities. This is because the metric takes into account the effects of time dilation and length contraction, which are fundamental principles of special relativity. So, while the time coordinate may appear different for