A weekend of relativity

docs/apps/FourierTransforms/app.py

@@ -71,8 +71,6 @@
 
 def server(input, output, session):
     click_data = reactive.value(None)
-    blue = 'navy'
-    red = 'firebrick'
 
     x_axis = np.linspace(-20, 20, 1024)
     freq_axis = np.fft.fftfreq(len(x_axis), d=(x_axis[1] - x_axis[0]))

docs/apps/LagrangianPoints/app.md

@@ -0,0 +1,16 @@
+---
+authors:
+  - ptuemmler
+categories:
+  - Physics
+tags:
+  - Draft
+date: 2025-11-23
+hide:
+  - toc
+---
+
+# Lagrangian Points
+<!-- more -->
+{{embed_app("100%", "830px")}}
+Underlying potential taken from: [https://doi.org/10.1086/381315](https://doi.org/10.1086/381315)

docs/apps/LagrangianPoints/app.py

@@ -0,0 +1,68 @@
+import numpy as np
+import plotly.graph_objects as go
+from shiny import App, Inputs, Outputs, Session, render, ui
+
+app_ui = ui.page_sidebar(
+    ui.sidebar(
+        ui.input_slider("q", "Mass Ratio (q)", min=0.0, max=1.0, value=0.05, step=0.01),
+        # ui.input_slider("x", "x", min=0.0, max=2.0, value=1.0, step=0.01),
+        ui.input_dark_mode(id='dark_mode'),
+    ),
+    ui.output_ui("plot")
+)
+
+def server(input: Inputs, output: Outputs, session: Session):
+    @render.ui
+    def plot():
+        # Set template based on dark mode
+        if input.dark_mode() == "dark":
+            template = "plotly_dark"
+        else:
+            template = "plotly_white"
+
+        fig = go.Figure()
+        # Add isosurface representing the potential field
+        x, y = np.meshgrid(np.linspace(-2, 2, 256),
+                           np.linspace(-2, 2, 256))
+        z = np.zeros_like(x)
+
+        q = input.q()
+        # x_pos = input.x()
+        values = (x - q/(1+q))**2 + y**2 + 2/((1+q)*np.sqrt(x**2 + y**2 + z**2)) + 2*q/((1+q)*np.sqrt((x-1)**2 + y**2 + z**2))
+
+
+        values -= np.min(values)
+        min_value = 1.5
+        values[values > min_value] = np.nan  # Mask values above a certain threshold
+
+        fig.add_trace(go.Surface(
+            contours = {
+                "z": {"show": True, "start": -min_value, "end": 0.01, "size": 0.05}
+            },
+            x=x,
+            y=y,
+            z=-values,  # Offset for better visibility
+            colorscale='turbo',
+            colorbar_title_text='f(x,y)',
+            showscale=True,
+            hoverinfo='skip'
+        ))
+
+        # Set layout
+        fig.update_layout(
+            template=template,
+            scene=dict(
+                xaxis_title='X-axis',
+                yaxis_title='Y-axis',
+                zaxis_title='Z-axis',
+                xaxis_showspikes=False,
+                yaxis_showspikes=False,
+                zaxis_showspikes=False,
+            ),
+            height=700,
+            margin=dict(l=0, r=0, t=0, b=0),
+        )
+
+        return ui.HTML(fig.to_html())
+
+app = App(app_ui, server)

docs/apps/LagrangianPoints/requirements.txt

@@ -0,0 +1,2 @@
+numpy
+plotly

docs/apps/MinkowskiSpaceTime/app.md

@@ -16,4 +16,6 @@ draft: true
 ---
 # Minkowski Space-Time
 <!-- more -->
-{{embed_app("100%", "800px")}}
+{{embed_app("100%", "800px", "stationary")}}
+{{embed_app("100%", "800px", "doppler")}}
+{{embed_app("100%", "800px", "motion")}}

docs/apps/MinkowskiSpaceTime/doppler/app.py

@@ -0,0 +1,151 @@
+import matplotlib.pyplot as plt
+from matplotlib.colors import LinearSegmentedColormap, Normalize
+import numpy as np
+import pandas as pd
+from shiny import App, render, ui, reactive
+
+berlin_cmap = [[0.621082, 0.690182, 0.999507],[0.612157, 0.689228, 0.995374],[0.603202, 0.688250, 0.991239],[0.594200, 0.687257, 0.987092],[0.585165, 0.686248, 0.982922],[0.576088, 0.685222, 0.978733],[0.566961, 0.684166, 0.974524],[0.557791, 0.683098, 0.970288],[0.548590, 0.681992, 0.966016],[0.539327, 0.680859, 0.961704],[0.530034, 0.679691, 0.957350],[0.520687, 0.678484, 0.952942],[0.511295, 0.677230, 0.948466],[0.501863, 0.675908, 0.943923],[0.492368, 0.674526, 0.939297],[0.482832, 0.673075, 0.934574],[0.473239, 0.671530, 0.929751],[0.463610, 0.669898, 0.924806],[0.453931, 0.668152, 0.919735],[0.444213, 0.666275, 0.914518],[0.434440, 0.664271, 0.909136],[0.424645, 0.662120, 0.903586],[0.414818, 0.659791, 0.897845],[0.404975, 0.657289, 0.891905],[0.395137, 0.654579, 0.885750],[0.385296, 0.651674, 0.879368],[0.375493, 0.648536, 0.872757],[0.365742, 0.645164, 0.865903],[0.356059, 0.641552, 0.858801],[0.346453, 0.637692, 0.851451],[0.336982, 0.633574, 0.843855],[0.327642, 0.629189, 0.836017],[0.318487, 0.624551, 0.827937],[0.309539, 0.619657, 0.819628],[0.300784, 0.614497, 0.811108],[0.292309, 0.609115, 0.802379],[0.284098, 0.603485, 0.793470],[0.276205, 0.597634, 0.784386],[0.268595, 0.591580, 0.775143],[0.261308, 0.585335, 0.765780],[0.254368, 0.578908, 0.756296],[0.247753, 0.572328, 0.746719],[0.241464, 0.565596, 0.737066],[0.235515, 0.558748, 0.727351],[0.229842, 0.551802, 0.717600],[0.224503, 0.544750, 0.707805],[0.219485, 0.537628, 0.697998],[0.214694, 0.530433, 0.688190],[0.210172, 0.523193, 0.678377],[0.205889, 0.515897, 0.668578],[0.201771, 0.508598, 0.658787],[0.197878, 0.501258, 0.649030],[0.194172, 0.493903, 0.639287],[0.190556, 0.486541, 0.629572],[0.187112, 0.479181, 0.619898],[0.183752, 0.471826, 0.610241],[0.180500, 0.464474, 0.600622],[0.177365, 0.457117, 0.591037],[0.174264, 0.449788, 0.581483],[0.171224, 0.442474, 0.571966],[0.168242, 0.435172, 0.562486],[0.165292, 0.427884, 0.553021],[0.162439, 0.420608, 0.543603],[0.159545, 0.413370, 0.534210],[0.156739, 0.406147, 0.524856],[0.153905, 0.398932, 0.515524],[0.151122, 0.391757, 0.506230],[0.148346, 0.384591, 0.496972],[0.145641, 0.377462, 0.487751],[0.142879, 0.370343, 0.478544],[0.140138, 0.363257, 0.469389],[0.137466, 0.356204, 0.460239],[0.134777, 0.349162, 0.451147],[0.132079, 0.342150, 0.442085],[0.129401, 0.335173, 0.433042],[0.126735, 0.328195, 0.424036],[0.124090, 0.321259, 0.415071],[0.121456, 0.314347, 0.406144],[0.118899, 0.307460, 0.397234],[0.116316, 0.300608, 0.388376],[0.113731, 0.293781, 0.379546],[0.111187, 0.286980, 0.370748],[0.108613, 0.280217, 0.362004],[0.106159, 0.273497, 0.353280],[0.103670, 0.266776, 0.344594],[0.101183, 0.260108, 0.335952],[0.098776, 0.253467, 0.327342],[0.096347, 0.246850, 0.318783],[0.094059, 0.240264, 0.310267],[0.091788, 0.233727, 0.301758],[0.089506, 0.227245, 0.293318],[0.087341, 0.220800, 0.284914],[0.085142, 0.214360, 0.276576],[0.083069, 0.207981, 0.268249],[0.081098, 0.201631, 0.259992],[0.079130, 0.195361, 0.251781],[0.077286, 0.189136, 0.243589],[0.075571, 0.182943, 0.235502],[0.073993, 0.176835, 0.227434],[0.072410, 0.170785, 0.219433],[0.071045, 0.164795, 0.211500],[0.069767, 0.158901, 0.203628],[0.068618, 0.153040, 0.195818],[0.067560, 0.147319, 0.188124],[0.066665, 0.141671, 0.180452],[0.065923, 0.136076, 0.172917],[0.065339, 0.130695, 0.165458],[0.064911, 0.125349, 0.158169],[0.064636, 0.120132, 0.150946],[0.064517, 0.115070, 0.143889],[0.064554, 0.110222, 0.136957],[0.064749, 0.105427, 0.130230],[0.065100, 0.100849, 0.123569],[0.065383, 0.096469, 0.117170],[0.065574, 0.092338, 0.111008],[0.065892, 0.088201, 0.104982],[0.066388, 0.084134, 0.099288],[0.067108, 0.080051, 0.093829],[0.068193, 0.076099, 0.088470],[0.069720, 0.072283, 0.083025],[0.071639, 0.068654, 0.077544],[0.073978, 0.065058, 0.072110],[0.076596, 0.061657, 0.066651],[0.079637, 0.058550, 0.061133],[0.082963, 0.055666, 0.055745],[0.086537, 0.052997, 0.050336],[0.090315, 0.050699, 0.045040],[0.094260, 0.048753, 0.039773],[0.098319, 0.047041, 0.034683],[0.102458, 0.045624, 0.030074],[0.106732, 0.044705, 0.026012],[0.110986, 0.043972, 0.022379],[0.115245, 0.043596, 0.019150],[0.119547, 0.043567, 0.016299],[0.123812, 0.043861, 0.013797],[0.128105, 0.044459, 0.011588],[0.132315, 0.045229, 0.009531],[0.136451, 0.046164, 0.007895],[0.140635, 0.047374, 0.006502],[0.144884, 0.048634, 0.005327],[0.149230, 0.049836, 0.004346],[0.153685, 0.050997, 0.003537],[0.158309, 0.052130, 0.002882],[0.163014, 0.053218, 0.002363],[0.167811, 0.054240, 0.001963],[0.172736, 0.055172, 0.001669],[0.177801, 0.056018, 0.001469],[0.182863, 0.056820, 0.001340],[0.188058, 0.057574, 0.001262],[0.193233, 0.058514, 0.001226],[0.198463, 0.059550, 0.001227],[0.203778, 0.060501, 0.001260],[0.209092, 0.061486, 0.001322],[0.214470, 0.062710, 0.001412],[0.219897, 0.063823, 0.001529],[0.225345, 0.065027, 0.001675],[0.230856, 0.066297, 0.001853],[0.236422, 0.067645, 0.002068],[0.242016, 0.069092, 0.002325],[0.247681, 0.070458, 0.002632],[0.253390, 0.071986, 0.002998],[0.259176, 0.073640, 0.003435],[0.264997, 0.075237, 0.003955],[0.270934, 0.076965, 0.004571],[0.276928, 0.078822, 0.005301],[0.283017, 0.080819, 0.006161],[0.289196, 0.082879, 0.007171],[0.295466, 0.085075, 0.008349],[0.301858, 0.087460, 0.009726],[0.308387, 0.089912, 0.011455],[0.315024, 0.092530, 0.013324],[0.321806, 0.095392, 0.015413],[0.328738, 0.098396, 0.017780],[0.335805, 0.101580, 0.020449],[0.343036, 0.104977, 0.023440],[0.350413, 0.108640, 0.026771],[0.357947, 0.112564, 0.030456],[0.365629, 0.116658, 0.034571],[0.373470, 0.120971, 0.039115],[0.381463, 0.125606, 0.043693],[0.389583, 0.130457, 0.048471],[0.397845, 0.135474, 0.053136],[0.406220, 0.140795, 0.057848],[0.414690, 0.146274, 0.062715],[0.423229, 0.151979, 0.067685],[0.431837, 0.157906, 0.073044],[0.440444, 0.164028, 0.078620],[0.449085, 0.170269, 0.084644],[0.457704, 0.176666, 0.090869],[0.466314, 0.183213, 0.097335],[0.474900, 0.189888, 0.104064],[0.483420, 0.196677, 0.111039],[0.491910, 0.203516, 0.118190],[0.500322, 0.210433, 0.125501],[0.508690, 0.217425, 0.132983],[0.516977, 0.224432, 0.140623],[0.525197, 0.231543, 0.148349],[0.533349, 0.238624, 0.156261],[0.541440, 0.245755, 0.164233],[0.549481, 0.252923, 0.172265],[0.557462, 0.260091, 0.180403],[0.565378, 0.267255, 0.188640],[0.573272, 0.274461, 0.196924],[0.581112, 0.281673, 0.205237],[0.588920, 0.288894, 0.213625],[0.596716, 0.296114, 0.222054],[0.604484, 0.303345, 0.230529],[0.612228, 0.310617, 0.239052],[0.619976, 0.317867, 0.247618],[0.627708, 0.325132, 0.256189],[0.635438, 0.332443, 0.264815],[0.643173, 0.339745, 0.273490],[0.650917, 0.347064, 0.282179],[0.658661, 0.354395, 0.290887],[0.666419, 0.361751, 0.299640],[0.674194, 0.369121, 0.308415],[0.681975, 0.376518, 0.317219],[0.689783, 0.383920, 0.326043],[0.697596, 0.391354, 0.334929],[0.705434, 0.398794, 0.343796],[0.713288, 0.406271, 0.352720],[0.721158, 0.413757, 0.361662],[0.729054, 0.421259, 0.370618],[0.736968, 0.428796, 0.379616],[0.744900, 0.436349, 0.388639],[0.752851, 0.443923, 0.397680],[0.760831, 0.451512, 0.406747],[0.768821, 0.459124, 0.415838],[0.776844, 0.466756, 0.424962],[0.784879, 0.474407, 0.434092],[0.792935, 0.482080, 0.443269],[0.801009, 0.489763, 0.452465],[0.809110, 0.497486, 0.461672],[0.817222, 0.505207, 0.470910],[0.825358, 0.512962, 0.480170],[0.833517, 0.520732, 0.489445],[0.841692, 0.528527, 0.498763],[0.849885, 0.536335, 0.508096],[0.858092, 0.544161, 0.517448],[0.866324, 0.552013, 0.526825],[0.874568, 0.559879, 0.536218],[0.882829, 0.567761, 0.545643],[0.891110, 0.575670, 0.555082],[0.899407, 0.583585, 0.564550],[0.907716, 0.591530, 0.574038],[0.916031, 0.599492, 0.583552],[0.924368, 0.607473, 0.593095],[0.932714, 0.615460, 0.602649],[0.941076, 0.623483, 0.612229],[0.949447, 0.631512, 0.621832],[0.957832, 0.639563, 0.631467],[0.966219, 0.647628, 0.641113],[0.974619, 0.655718, 0.650792],[0.983030, 0.663823, 0.660487],[0.991448, 0.671939, 0.670216],[0.999873, 0.680072, 0.679950]]
+#Taken from F. Crameri's scientific-colour-maps version 8.0.1.  https://doi.org/10.5281/zenodo.1243862 (included by default in newer matplotlib versions)
+berlin = LinearSegmentedColormap.from_list('berlin', berlin_cmap, N=256)
+max_iterations = 1e6
+
+
+app_ui = ui.page_sidebar(
+    ui.sidebar(
+        ui.input_slider(
+            "velocity",
+            "Velocity (units of c)",
+            min=-0.9,
+            max=0.9,
+            value=0.5,
+            step=0.01,
+            animate=True
+        ),
+        ui.input_slider("period", "Period of signal (self time)", min=1, max=3, value=2, step=0.1, animate=True),
+        ui.input_radio_buttons("signal_type", "Signal type", choices={"ping": "ping", "periodic": "periodic"}, selected="periodic", inline=True),
+        ui.input_dark_mode(id='dark_mode'),
+        open='always'
+    ),
+    ui.output_plot(
+        "plot",
+        click=True,
+        width="100%", height="700px"
+    ),
+)
+
+
+def lorentz_transform(x, t, v):
+    c = 1 # velocity of light v will be in units of c
+    gamma = 1 / (1 - (v**2 / c**2))**0.5
+    x_prime = gamma * (x + v * t)
+    t_prime = gamma * (t + (v * x) / c**2)
+    return x_prime, t_prime
+
+
+def mirrored(maxval, inc=1):
+    x = np.arange(inc, maxval, inc)
+    return np.r_[-x[::-1], 0, x]
+
+
+def server(input, output, session):
+    tlimits = (-10, 10)
+    points_per_period = 12
+    color_norm = Normalize(vmin=-1, vmax=1)
+
+    @render.plot()
+    def plot():
+        if input.dark_mode() == "dark":
+            style_label = 'dark_background'
+            blue = 'lightsteelblue'
+            red = 'lightcoral'
+            cmap = berlin
+        else:
+            style_label = 'seaborn-v0_8'
+            blue = 'navy'
+            red = 'firebrick'
+            cmap = plt.get_cmap('RdBu_r')
+
+        with plt.style.context(style_label):
+            fig, ax = plt.subplot_mosaic([['minkowski', 'sender'], ['minkowski', 'receiver']])
+
+            v = input.velocity()
+            period = input.period()
+
+            signal_times = mirrored(tlimits[1], inc=period)
+            if input.signal_type() == 'ping':
+                signal_amplitude = np.ones_like(signal_times)
+            else: # input.signal_type() == 'periodic':
+                signal_times = np.linspace(signal_times[0], signal_times[-1], points_per_period * (len(signal_times) - 1) + 1)
+                signal_amplitude = np.cos(2 * np.pi * signal_times / period)
+
+            for indx, signal in enumerate(signal_times):
+
+                if input.signal_type() == 'ping':
+                    color = 'grey'
+                else: # input.signal_type() == 'periodic':
+                    # Color line according to amplitude by mapping to colormap between -1 and 1
+                    color = cmap(color_norm(signal_amplitude[indx]))
+
+                ax['sender'].plot([signal, signal], [0, signal_amplitude[indx]], color=color)
+
+                ct_axis_transformed, x_axis_transformed = lorentz_transform(signal, 0, v)
+
+                sign = np.sign(signal) * np.sign(v)
+
+                intersection = ct_axis_transformed + sign * x_axis_transformed
+
+                # Extend signal to receiver worldline with slope 1
+                ax['minkowski'].plot([x_axis_transformed , 0], [ct_axis_transformed, intersection], color=color)
+                ax['receiver'].plot([intersection, intersection], [0, signal_amplitude[indx]], color=color)
+
+
+            ax['minkowski'].axvline(0, color=blue, label='Reciever worldline')
+            ax['minkowski'].axline((0, 0), slope=np.tan(np.pi/2 - np.atan(v)), color=red, label='Sender worldline')
+
+            ax['sender'].axhline(0, color=red)
+            ax['sender'].axvline(0, color=red)
+            ax['receiver'].axhline(0, color=blue)
+            ax['receiver'].axvline(0, color=blue)
+
+            ax['sender'].set_title("Sender Frame (in motion)")
+            ax['receiver'].set_title("Receiver Frame (at rest)")
+            ax['minkowski'].set_title("Minkowski Spacetime Diagram")
+
+            ax['sender'].set_xlabel("sender self time in seconds")
+            ax['sender'].set_ylabel("signal amplitude")
+            ax['receiver'].set_xlabel("receiver self time in seconds")
+            ax['receiver'].set_ylabel("signal amplitude")
+            #
+            ax['minkowski'].set_xlabel("x in light-seconds")
+            ax['minkowski'].set_ylabel("ct in light-seconds")
+            ax['minkowski'].set_aspect('equal')
+
+            # # Add self time ticks according to x lims of sender and receiver plots
+            # sender_xticks = ax['sender'].get_xticks()
+            # for time in sender_xticks:
+            #     sign = np.sign(time) * np.sign(v)
+            #     ct_axis_transformed, x_axis_transformed = lorentz_transform(time, 0, v)
+            #     if sign < 0:
+            #         ax['minkowski'].text(x_axis_transformed, ct_axis_transformed, f'τ={time:.0f}s    ', color=red, verticalalignment='center',
+            #             horizontalalignment='right', )
+            #     else:
+            #         ax['minkowski'].text(x_axis_transformed, ct_axis_transformed, f'    τ={time:.0f}s', color=red, verticalalignment='center',
+            #             horizontalalignment='left')
+            #
+            # receiver_xticks = ax['receiver'].get_xticks()
+            # for time in receiver_xticks:
+            #     sign = np.sign(time) * np.sign(v)
+            #     if sign > 0:
+            #         ax['minkowski'].text(0, time, f'τ={time:.0f}s    ', color=blue, verticalalignment='center',
+            #                              horizontalalignment='right', )
+            #     else:
+            #         ax['minkowski'].text(0, time, f'    τ={time:.0f}s', color=blue, verticalalignment='center',
+            #                              horizontalalignment='left')
+
+            xlims = ax['minkowski'].get_xlim()
+            new_lim = max(5, xlims[1])
+            ax['minkowski'].set_xlim(-new_lim, new_lim)
+        return fig
+
+
+app = App(app_ui, server, debug=True)