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60 views

Lorentz Factor from Minkowski's Original Paper 'Space and Time'

Consider the following figure: Minkowski, in his paper 'Space and Time', derives the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ from considerations of this figure. He ...
4
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2answers
87 views

Lorentz-transformation

I don't understand how to derive the matrix representing the Lorentz-transformation given two systems S and S': $$x' = \Lambda x$$ these transformations do not leave the differences $\Delta x^\mu$ ...
1
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1answer
85 views

Questions about special relativity, index in the Lorentz matrix

I'm studying special relativity I have read this: We have $ x^u = (ct, x^1,x^2,x^3) $. If we apply Lorentz transformation we can write: $x'^u = \Lambda^{u}_{\hspace{0,2 cm}\nu} x^{\nu} $ $x'_u =...
1
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1answer
111 views

Intuitive explanation for the Lorentz transformation for time

I've recently started learning SR, and while the Lorentz transformation for space is pretty obvious, just the Galilean transformation combined with space contraction, I can't figure out the ...
5
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2answers
88 views

Spacetime diagrams and their interpretation

Recently started an introductory course of relativity, and started learning about space time diagrams. I couldn’t figure out what are the uses of a spacetime diagram as an alternative to Lorentz ...
2
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1answer
87 views

How do time-like separated points preserve temporal ordering under orthochronous Lorentz Transformations?

How do time-like separated points preserve temporal ordering under orthochronous Lorentz Transformations? This question has already been asked in this Phys.SE post but I want to derive this result in ...
0
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1answer
67 views

How does the Equivalence Principle imply that derivatives of the metric vanish in a freely falling frame?

Why do the first derivatives of $g_{\mu\nu}$ vanish in a freely falling coordinate system? I would like to start from the Equivalence Principle that for any point in spacetime there exists a locally ...
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0answers
42 views

Proving two space-time intervals are equivalent with matrix algebra

η=$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $ v=$ \begin{bmatrix} ct\\ x\\ y\\ z \end{...
0
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2answers
138 views

Spacelike and timelike intervals confusion

I'm confused about this, specifically the spacetime interval. A timelike interval is one in which 2 events can be related to each other in a given reference frame within its light cone, that is, it ...
1
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2answers
98 views

Paradox are all 4D distances zero?

The 4 dimension distance from the origin of a point is $\sqrt{x^2+y^2+z^2-t^2}$. Which means the 4 dimensional distance on the light-cone is zero. Take a point A and a point B in the future at ...
0
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4answers
179 views

Invariance of the relativistic interval

Imagine we have two events, $E_1, E_2$ in the coordinate systems $K, K'$ (with coordinates $(x,y,z,t),\ (x',y',z',t')$), whilst $K'$ ist moving with the speed $\vec v$ in regard to $K$. From the ...
1
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2answers
96 views

Minimum separation from the spacetime interval

I've been working through invariant spacetime interval questions recently, and I came across a question in my lecture notes where; $$\Delta s^2=\Delta x^2 -(c\Delta t)^2 > 0 $$ Now it is clear to ...
0
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1answer
192 views

The derivation of the Lorentz transformation: addition of distances

In the derivation of the Lorentz transformation, one has a reference frame, $S$, at rest and another, $S'$, moving away at constant speed $v$. At time $t$ there is an event at a point $x$ in $S$. The ...
4
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1answer
125 views

Invariance of spacetime interval in special relativity: linearity

I'm trying to understand which assumption are necessary to prove the invariance of the spacetime interval $$\Delta s^2=c^2\Delta t^2-\Delta \mathbf{x}^2$$ in special relativity. The postulates of ...
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2answers
75 views

Clearing up a discrepancy when deriving the Lorentz transformation from length contraction

I've been working through the Feynman Lectures on Physics. I'm currently on lecture 15: The Special Theory of Relativity, specifically 15-5, the section on the deriving the Lorentz Transformation ...
2
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3answers
129 views

Lorentz Velocity Transform With Tensor Notation [closed]

So I'm attempting to prove the Lorentz Velocity tranform: $${v_x}' =\frac{v_x-u}{1-v_xu/c^2} $$ using tensor notation. In this case obviously $\beta = u/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The ...
0
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1answer
264 views

Proving electromagnetic wave equation is Lorentz invariant

I am trying to prove that the electromagnetic wave equation is invariant under Lorentz transformation. I need to show that $$\frac{d^2U}{dx^2}-\frac{1}{c^2}\frac{d^2U}{dt^2}=\frac{d^2U}{dx'^2}-\frac{...
1
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1answer
356 views

Invariance of spacetime interval directly from postulate

In Special Relativity, the spacetime interval $$\mbox{d}s^2 = \mbox{d}t^2 - \mbox{d}x^2 - \mbox{d}y^2 - \mbox{d}z^2 \tag{$\star$}$$ between two events is well known to be invariant under Lorentz ...
3
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2answers
82 views

Proof for uniqueness of transformation between relativistic frames

My understanding of the Lorentz transformation is that to ensure that laws of physics remain frame-independent, a transformation was devised, which we call today by the name Lorentz Transformation. ...
1
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1answer
129 views

Deriving Lorentz transformation

So I have been trying to derive the Lorentz transormation via the equation of a photon: $$\frac {\partial^2 \psi}{\partial t^2}-c^2 \frac{\partial^2 \psi}{\partial x^2}=0$$ The premise is that this ...
0
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2answers
428 views

The matrix of the Lorentz transformation is or isn't a tensor?

The matrix of the Lorentz transformation isn't a tensor, because it switches the sign of the non-diagonal components during the inverse transformation, right? So it isn't 'basis independent', but the ...
1
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3answers
90 views

Interval Preserving in Minkowski Space

The squared line element in any spacetime is given as $$ds^{2}=g_{ab}dx^{a}dx^{b}.$$ The use of tensors helps us to infer that the line element in some other frame would be $$ds'^{2}=g'_{ab}dx'^{a}dx'^...
2
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4answers
830 views

Hyperbolic rotation of spacetime and Lorentz transformation

My question is: What is the motivation behind deriving Lorentz transformation using hyperbolic functions? Is it because the formulation in such way offers a handy mathematical tool? Or is there ...
1
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2answers
679 views

How to derive the spacetime interval from the Lorentz transformation?

I've seen plenty of derivations of the Lorentz transformation from the spacetime interval. Can this process be reversed, to derive the spacetime interval from the Lorentz transformation and the two ...
1
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0answers
60 views

What is the physical significance of locally Minkowskian but not necessarily geodesic coordinates?

Consider an event $P$ in spacetime and a coordinate system $x_\mu$ such that at $P$: $g_{\mu\nu}=\eta_{\mu\nu}$ $g_{\mu\nu,\sigma}=0$, or equivalently the Christoffel symbols vanish at $P$. $x_\mu$ ...
1
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3answers
424 views

Pure Lorentz boost; transpose $\neq$ inverse?

By definition a matrix representing a Lorentz transformation is orthogonal, so that its inverse is equal to its transpose. Consider a pure boost in the t-x plane; $$\Lambda_x=\begin{pmatrix} \cosh(\...
2
votes
1answer
99 views

How did we get the time dilation equation from this proper time equation?

As you can see in the text book we get time dilation equation from equation number 1.9 I just can't see how he did it here: For now, we define the proper time between events B and the origin to be ...
0
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2answers
100 views

I don't understand the meaning of $M_{\alpha\beta}$ in this equation

I don't understand the meaning of $M_{\alpha\beta}$ here in this equation, is it a matrix, or it is a function of a variable ?
1
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1answer
303 views

Proof of conservation of spacetime interval [duplicate]

In this paper as well as in Landau's textbook on classical field theory, there is a proof of the conservation of spacetime interval in which authors deduce, that the differentials of the interval in ...
2
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2answers
3k views

Why is spacetime interval invariant under Lorentz transform?

We can verify that the spacetime interval is invariant through brute force computation. Is there a deeper reason why the interval is invariant under Lorentz transform?
0
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1answer
96 views

How do I derive Lorentz transformations from $S^2=c^2t_1^{\,2}-v_1^{\,2}t_1^{\,2}=c^2t_2^{\,2}-v_2^{\,2}t_2^{\,2}$

As in the picture, a typical vertical light clock is moving to the right at $v_1$ for one observer and at $v_2$ for the other observer. How do we derive full Lorentz transformations from this?
2
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3answers
166 views

Underlying structure behind relativity?

Is there a way to understand the underlying "cause" for observing (special) relativity? What is the structure/geometry that makes these observations make sense? I noticed that if all objects are ...
3
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1answer
260 views

Is the Euclidean metric the only one invariant under Galilean Transformations?

Is $$ds^2=dx^2+dy^2+dz^2$$ the only metric that is invariant under Galilean transformations? And if yes how do you prove it?
0
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1answer
123 views

Special relativity in layman's terms

So I'm learning special relativity in high school right now, and I'm having some difficulty understanding some of the relations. Right now, we're learning about time dilation and length contraction....
3
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4answers
1k views

Postulate implies existence Lorentz transformation?

My textbook about Special Relativity says that the existence of Lorentz Transformation is guaranteed by the postulates of Special Relativity. So, I'm assuming it's the first postulate we're talking ...
8
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2answers
443 views

Existence proof of Lorentz transformation from lightlike to lightlike vectors

This is a question that might have quiet an easy answer and intuitively has an easy solution, but I struggle a bit with the strict mathematical proof. The statement is relatively simple: Let $t^\mu,...
9
votes
2answers
783 views

Homogeneity and isotropy and derivation of the Lorentz transformations

In deriving the Lorentz transformations I have found (from reading a few different sets lecture notes) that it is argued that they must be linear and thus there general form must be $$x'=Ax+Bt,\quad t'...
0
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0answers
37 views

Inertial coordinate systems [duplicate]

In Newtonian mechanics, by the following two assumptions: (i) The time is absolute. (ii) The length is absolute. it is easy find the relations betweem two coordinate systems with uniform motion ...
1
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2answers
208 views

Definition of the Lorentz transformations [closed]

Until very recently I believed that the Lorentz transformations were defined as "the transformations that carry one inertial reference frame into another". In Wikipedia's page we find something along ...
11
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3answers
620 views

Can special relativity be derived from the invariance of the interval?

As far as I know, the classical approach to special relativity is to take Einstein's postulates as the starting point of the logical sequence, then to derive the Lorentz transformations from them, and ...
2
votes
2answers
2k views

Schutz's geometrical proof that spacetime interval is invariant

I'm trying to understand the proof that spacetime interval is invariant under for any two inertial observers. I know it's easy to arrive at the result using Lorentz transformation but I'm trying to ...
3
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1answer
185 views

How to motivate this approach on Special Relativity?

One of the most common approaches to Special Relativity is that based on the two Einstein's postulates: The laws of Physics are invariant in every inertial reference frame The speed of light ...
1
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3answers
901 views

Spacelike to timelike four vectors

First at all, let me just say that I'm not a Physicist, I study mathematics. So, I have this question. If you have a spacelike four vector, is there any transformation that could change it to be a ...
4
votes
4answers
3k views

Are Lorentz transformations linear transformations? [duplicate]

My textbook says that Lorentz transformations are linear transformations and present them as matrices. Lorentz transformations relate different coordinate systems with each other. It seems that ...
5
votes
1answer
1k views

Minkowski metric — why does it follow from the constancy of the speed of light? [duplicate]

In all the sources I’ve been able to find, the Minkowski metric appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates ...
7
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3answers
2k views

Homogeneity of space implies linearity of Lorentz transformations

In the derivation of Lorentz transformations, the Wikipedia article mentions a couple of times that the linearity comes from the homogeneity of space. I am looking for a thorough explanation on this.
15
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2answers
5k views

How do I derive the Lorentz contraction from the invariant interval?

Reviewing some basic special relativity, and I stumbled upon this problem: From the definition of the proper time: $$c^2d\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using $...
7
votes
4answers
3k views

Why is time-order invariant in timelike interval?

Why do two observers measure the same order of events if we are inside the light cone? (e.g. if $ds^2 > 0$ time-order is preserved according to the classical mechanics book I am reading, but it ...
9
votes
4answers
7k views

Deriving the Lorentz Transformation

I have been trying to understand a more or less geometric derivation of the Lorentz transformation, and I'm getting stuck at one spot. The wikipedia article for the Lorentz transformation for frames ...